This page associated to the Poncelet–Steiner theorem describes a number of Steiner constructions and related insights and alternative constructions, that are of unique interest for those fascinated by the Poncelet–Steiner theorem and its variants (such as portions of the arc, omission of circle centers, etc.), including the inline references to constructions with parallelograms, squares, etc., as well as other alternative constructions. This page exists to elaborate on the discussions of the Poncelet-Steiner theorem, and develop projective geometry tools to assist in the universality, generality, convenience, and insights of that theorem.
The constructions described below are ordered such that if another construction is required, it is prior listed. No construction requires another listed later, though some may reference constructions found in the Poncelet-Steiner theorem main article. There may also be passive references to other tertiary constructions reserved for yet another subpage. It is assumed that the reader/geometer has read and understood the main article, and has a grasp of the constructions outlined there.
In this section we revisit and generalize two constructions found in the Poncelet-Steiner Theorem main article. The uses for the generalization will be justified in later sections, and the particular cases are demonstrated in this section.
Given a line with segment AB, and a point C (or D) on the line, we may construct the projective harmonic conjugate, point D (or C). The points C and D are each projective harmonic conjugates of one another with respect to, and defined by, the segment AB. Indeed, it is also true that points A and B are each projective harmonic conjugates of one another with respect to the segment CD, though in this case the latter point pair was determined from the former.
The projective harmonic conjugates define a proportionality with the segment AB. It is the case that .
The conjugates may be constructed thus:
Alternatively, point C may be constructed from point D using the same set of lines, in which point Q is found before point X.
The parallel line construction from a bisected line-segment, as seen in the Poncelet-Steiner Theorem article, is a special case of the Projective Harmonic Conjugate construction. In this case, because one of the key defining points of the harmonic is a midpoint, point C, the conjugate point D exists "at infinity" (and thus fails to exist on the line), thereby compelling the line PQ to be parallel. As such, there is no point D on line AB that represents the points of conjugacy. In this construction, however, point D is not the construction of interest, but rather its associated line, the (blue) line PQ.
The construction of the midpoint from two parallels, as seen in the Poncelet-Steiner Theorem article, is a special case of the Projective Harmonic Conjugate construction. In this case, because one of the key defining points, D, of the harmonic does not exist (or rather exists "at infinity"), due to the two lines being parallel, its conjugate placed interior to the segment AB, point C, must be the midpoint. The midpoint construction involves the same set of lines and points as the parallel line construction or the harmonic conjugate construction, simply in a different order and different set of initial conditions.
Given any two lines, m and n, a concurrent line through a point, P, is a line that intersects both lines m and n at the same point that lines m and n intersect one another. In other words, if lines m and n intersect at an unconstructed point Z (off the page, not visible, at great distance), then the concurrent line is the line . Concurrent lines all intersect at the same point; from any two lines, concurrent lines may be constructed through any point in the plane.
The construction is convenient when a third line is required, defined by a point P and the (possibly unconstructed) intersection point between two other lines. This is a particularly useful construction when the point of intersection of the two lines is off the page at a great distance. The circle in the plane is not required.
Given lines m and n (both in black), and a point P through which the concurrent is to be constructed:
It is important to note that whenever point P is equidistant between the two lines m and n (i.e. existing on the angle bisector between them), the concurrent line construction will fail. Alternatively phrased, the construction will fail if the point P is the midpoint of any segment colinear with the transversal line, in which consecutive interior angles are congruent, and the endpoints of the segment exist on the two lines m and n. This is a special case scenario.
A concurrent through P may still be constructed, however. A simple solution is to construct an intermediary concurrent line first.
Given any two lines m and n, and an arbitrary point P in the plane, the concurrent of the lines may be constructed through the point using the projective harmonic conjugate. This construction does not require a circle, and avoids the special case scenario of the projective point method.
Given two parallel lines, m and n, a third parallel through an arbitrary point, P, can be constructed without the use of the circle.
This construction is merely a special case of the concurrent line construction, above, wherein the two given lines are already parallel. Thus, refer to the construction of the concurrent line in the previous section. It will not be repeated here. The parallel nature of lines m and n do not prohibit the concurrent line construction; the same set of constructable lines is achievable.
The principle behind this construction is that parallel lines "intersect at infinity", and concurrent lines intersect at the same point, therefore the concurrent to two parallels is also parallel.
Thus a third parallel through an arbitrary point may be constructed without a circle whenever there are already two parallels. The construction is trivial:
If the point P exists equidistant between the two parallel lines, the same special case is applicable and the construction will fail, if the projective point method is used; use an intermediary parallel. If the harmonic conjugate method is used, this should not be an issue.
Given are three arbitrary lines in space - lines a, b, and c - each parallel to one another, and the line in the middle is equidistant from the other two (i.e. "evenly spaced").
We may find the midpoint of any arbitrary line segment in space, PQ:
In an alternative construction, a midpoint of a line segment may be constructed from a parallelogram. The parallelogram allows for the construction of a line parallel to PQ, upon which the midpoint construction found in the Poncelet-Steiner theorem main article may be utilized.
Whenever three parallels of this type exist in the plane - with one line equidistant between the other two - any arbitrary line cutting through them can have a parallel of it constructed arbitrarily in space, as per steps 1 and 2 in the previous construction, because the three parallels intersect the cutting line such that a bisected segment exists. In the previous construction, points J, L, and S, form three adjacent of the four vertices of a parallelogram, the fourth being unlabeled. Thus, parallelograms are implied, and are arbitrarily constructable anywhere along the three parallels, such that any two of the parallels are opposing sides of the parallelogram. Any constructions involving the parallelogram are available.
Parallelograms are a useful tool in their own right, though not as powerful as a circle.
Given a line m and a parallelogram STUV (both in black), a second parallel can be constructed symmetrically across the parallelogram:
The principle is that parallels induce reflections. The parallelogram is not strictly speaking necessary; what is necessary is that there are two parallels intersecting the line being reflected, and an arbitrary point (X, in this case) that exists equidistant between the two parallels, across which we make reflections. The parallelogram allows for the construction of both the center point and the parallels, and because there are two distinct sets of parallels in a parallelogram we are assured the entire plane is covered.
Alternatively, line segment SU is bisected by point X. Two parallels may be made through points T and V, as per the parallel line constructions found on the Poncelet-Steiner theorem page. These parallels will intersect the given line m, and together with point X a reflection can be made.
Regarding the above construction involving the parallelogram, the parallelogram predetermines the location of the second parallel, as a reflection across the parallelogram. However, once a second parallel is in the plane, a third parallel may be constructed arbitrarily. Whenever a parallelogram exists in the plane, any line may have a parallel constructed from it, through any arbitrary point in the plane.
Two previous constructions are trivially combined:
Given any parallelogram ABCD, three equally spaced parallels may be constructed. Indeed, four distinct sets may be constructed.
Therefore, any constructions based on the existence of three equidistant parallels in the plane are achievable, whenever a parallelogram exists.
Since three equidistant parallels imply a parallelogram, and a parallelogram implies three equidistant parallels, they are equivalent premises in any construction.
Given is a circle, centered at point O, we may construct an arbitrary inscribed parallelogram:
Any constructions involving the parallelogram may be achieved when a circle is provided.
The parallelogram does not imply the circle. The circle (with its center) is a stronger proposition.
Indeed, a circle implies a square, which is stronger than the parallelogram. Any line passing through the center of the circle may have a parallel of it constructed also through the circle. From the intersection points of these lines with the circle, an isosceles trapezoid (or triangle) is defined. The non-parallel sides of the trapezoid may be extended to an intersection point, which along with the circle center defines a perpendicular. The perpendicular intersects the circle at two points. The two perpendicular lines passing through the circles center define the diagonals of a square; the vertices are the points where these lines intersect the circle. Refer to the later construction Constructing a Centerline from Two Cutting Parallels for some details.
The translation of a line segment is a relatively trivial process, and can be done with either a circle or a parallelogram, or by any other means that allows for the construction of parallel lines.
Given is line segment AB (in black), and a point A' to which the endpoint A is to be translated.
This construction will, however, fail if point A' is colinear with line AB. A simple solution for this condition is to translate the line segment AB to an intermediary non-colinear point X first, then translate it back to point A' .
The line segment rotation is another useful transformation, and is of particular use in Steiner constructions, as some constructions will fail under particular scenarios involving the coincidental orientation of line segments. The construction of the radical axis and the intersection between a line and a circle can suffer this scenario.
The Poncelet-Steiner theorem article covers one rotation method, which unfortunately fails when rotating a line segment a full 180 degrees. There is a workaround as explained in the article, but is unnecessarily burdensome. The construction provided below is an alternative approach to line segment rotation that is no more complicated than the one in the main article, and will work for half-circle rotations. This construction, however, has its own fallbacks.
Given is a line segment AB (in black), and an endpoint A about which the rotation is made. This may represent a circle A(B). Provided also is given circle O(r).
As mentioned, if point P is colinear with line AO this construction will fail. We may instead refer to the construction in the Poncelet-Steiner theorem article, or use a sequence of partial rotations to achieve the same ends, as was alternatively done in the main article.
Given is a line segment AB (in black), on and extended by line AB (in red), and two arbitrary parallel lines m and n (both in orange) in the plane. One or both parallels may be constructed if necessary. We may construct on line AB an extension of the line segment to any arbitrary integer multiple. In the image below, segment AC is two times, segment AD is three times, and segment AE is four times the length of segment AB.
Observe that on line m, a sequence of extensions are being generated, corresponding to those on line AB. This second set of segment extensions are scaled down in size, proportional with the distances between the parallel lines. For the purposes of this construction, the proportionality, and thus the parallel line spacing is arbitrary. Specifically, the ratio between segments AB and ab is equal to the ratio of the distance line AB is to line n and the distance line m is to line n. This feature could be intentionally utilized if one has control over the line spacing.
The construction of the secondary segment extension, on line m, is essential for the following section on segment sectioning.
Extensions may also be achieved through a series of colinear translations of the segment, as seen in a previous section, which will require the construction of arbitrary parallels. This construction, though it makes use of parallels, they are not strictly arbitrary and are presumed to already be present.
Given is a segment AB, which we wish to n-sect. In the example below, a trisection is performed. We are also given a parallel line m on which there already exists a trisected segment ad, which was constructed as a three-times extension of segment ab, and as a by product of a three-times extension of segment AB, as seen in the previous construction segment extensions.
Given is a line m (in black) across which reflections are to be made, the point P which we aim to reflect across the line, and the given circle (in green) centered at point O.
If line m passes through circle center O.
To reflect a single point about a line, the following construction is used:
To reflect any figure about a line - be they other lines, circles, or other figures - simply reflect each of its defining points about the line.
The construction is a trivial application of the point reflection:
Given is a square ABDC (in dark green) and an arbitrary line m (in black), and a point in the plane P. We wish to construct a perpendicular of line m through point P.
These constructions concern the construction of tangent lines, points of tangency, pole point, and the polar line, which runs through the points of tangency.
Given is an arbitrary circle (in green) with a center point O (which will not be utilized), and a point P existing on the arc of the circle. We wish to construct the tangent lines to the circle at the point. Such a point has a tangent line that is also the polar.
There are numerous properties involving the polar of which we may take advantage. Several different constructions avail themselves, and are useful under different circumstances.
These constructions work whether point P is internal or external to the circle, but will fail if it is on the arc of the circle or coincident with the circle center. In the former case, refer to the previous construction for a tangent line through a point on the circle. In the latter case, no polar exists.
Given is an arbitrary circle (in green) with a center point O (which will not be utilized), and a point P existing not on the circle. We wish to construct the tangent lines to the circle which pass through the point. Observe the relationships between the points of tangency on the circle, tangent lines through point P, the polar line of point P, and the inverse in the circle of point P.
It is valid to point out that, strictly speaking, points E and F and the line passing through them never needed to be drawn. Points A, B, C and D were sufficient. The point Y may be constructed as the intersection of lines AC and BD, although this is often less convenient. Additionally, point Y need not be constructed; we need only the concurrent between lines AC and BD passing through point X; this is the polar in the circle of point P.
Alternatively, we could have used points A and B (which exist on the same line through point P). From these points we could have constructed tangents to the circle, as per the previous construction for a tangent through a point on the arc of the circle. These tangent lines would intersect one another on the polar at a point X. We can then find point Y, also on the polar, by finding the intersection of the tangents to the circle at points C and D (both of which are colinear with point P). As before, concurrent line constructions may be employed when necessary.
We may also find a point X as the harmonic conjugate of point P on the line AB. Point Y may be found as the harmonic conjugate of point P on the line CD. The projective harmonic conjugate of point P on any cutting line through the circle exists on the polar. It should be noted that any chord through the circle on the line passing through point P defines a line segment upon which the projective harmonic conjugate of the point lies on the polar. Using multiple such chords, multiple points on the polar may be constructed, thus defining the line.
This is a relatively trivial construction. Let point P exist in the plane other than the center of the circle, point O. For point O no inverse exists.
For any circle with radius r and center point O, any point P in the plane and its inverse (point I) in the circle share a proportionality relationship. It is the case that .
Let also a line exist through point P that is centerline through the circle (that is, it passes through the center of the circle). The center point of the circle need not be constructed or provided. Naturally if the centerline does not exist, it must be constructed, either directly as the line through the circle center or through some other means.
The inverse of point P may be constructed simply.
If point P exists on the arc of the circle, nothing need be done. The point is its own inverse.
Let point P exist in the plane internal or external to the circle:
Alternatively, if the point P is internal the circle, a line through it perpendicular to the centerline may be constructed. Where this perpendicular intersects the circle, tangents to the circle may be constructed, which intersect one another at the inverse point. The line perpendicular to the centerline through point P may be constructed as the third parallel to the two tangents to the circle at the points the centerline intersect the circle.
Given the polar construction of the previous section using the projective harmonic conjugate technique, if the chord is chosen to be the diameter of the circle, the constructed conjugate is also the inverse of a point P in a circle.
Given an arbitrary line m in space and a circle (devoid of its center), we may find the point M that is the pole to the polar m in the circle.
This construction will work for polar lines that are both intersecting and non-intersecting to the circle.
This construction is fully general. Each of the special cases to follow are simplified special cases of this construction.
This is a trivial case. If the polar is tangential to the circle, the point of intersection to the circle is the pole. The pole is on the polar at the point of tangency. There is nothing to construct, but to find the point of tangency by intersecting the circle with the tangent.
For any polar intersecting the circle in two points A and B, we may construct the tangents to the circles at the points A and B. These tangent lines intersect one another at a unique point exterior to the circle. The intersection point is the pole.
If the polar is also a centerline (thus crossing the circle center) then there is no pole; or it "exists at infinity". The tangent lines to the circle, in this case, will be parallel.
Given are two parallel lines and a circle placed arbitrarily in the plane. Construct additional parallel lines as necessary, such that no fewer than two intersect the circle, with a total of four distinct intersection points between them.
Let one of the parallel lines intersect the circle at points A and B; let a second parallel intersect at points C and D. A centerline to the circle may be constructed simply:
Observe that a single midpoint construction (or segment bisection) may be used. Line segments AB and CD may be bisected concurrently, and bisected by the very centerline we wish to construct. The details may be found in the appropriate section on the Poncelet-Steiner theorem main article.
More concisely, we may define points X and Y by the intersections of lines AC with BD and AD with BC, respectively. The line XY is the desired centerline (which bisects both AB and CD concurrently).
Should a point (e.g. X) not exist due to two lines (e.g. AC with BD) being parallel, we may use concurrent line constructions, or we may use the two parallel lines AB and CD to construct a third parallel through the circle at a more convenient location for this construction. Alternatively, we may recognize that this scenario can only happen if ABCD is a rectangle, in which case each diagonal defines a centerline.
If one of the parallels happens to pass through the circle center, its intersection points with the circle defines two opposing vertices of a square inscribed in the circle. The constructed centerline, which is orthogonal and also passes through the circle center, defines the other two vertices. Thus a square may be defined, as per a previous section Circles Imply Squares.
Alternatively, we may find the two distinct poles corresponding to the two distinct parallel lines (treated as polars) to the circle. The two poles of these parallels define a perpendicular line which passes through the circle center.
Given two non-intersecting circles (both in dark green), each devoid of their centers, and given an arbitrary point on the centerline, point M, exterior to at least one of the two circles, it is possible to construct the centerline.
If the point M is in the interior of one of the circles, only a slight modification is necessary. The polar through only one of the circles is truly necessary. Tangent lines through the point M to any circle they can be constructed on will intersect the other circle at two points for each tangent line; they will be symmetric across the centerline. In the general case at least two additional centerline points may be constructed as intersections.
This construction will fail if the point M is interior to both circles simultaneously, one interior to the other but not intersecting. It will also fail if the point is the common point of tangency between the two circles, be they interior or exterior.
In general, from any point on the centerline, two parallel lines may be constructed which are perpendicular to the centerline. From this scenario, refer back to the centerline construction from two parallel lines in the plane.
When two arbitrary non-intersecting circles are provided - left and right circles l and r - without their centers, and a point P exists in the plain on the radical axis between the circles, the common centerline through the circles may be constructed.
This is a relatively trivial construction. The intersection point of the polars to each circle from the point on the radical axis, is also on the radical axis, and thus define the line:
Given is an arc of a circle (in dark green), of arbitrary arc length, spanning between the endpoints A and B of the arc AB. Given also is a line m (in black). We wish to intersect the line with the circle of the arc; that is to say, we wish to find the intersection points of the line with the circle which is defined by the arc.
Notice that for this construction the center, point O, of said circle is not utilized. It need not be provided at all in order to intersect a line with the arc of the circle. However, according to Steiner's theorem, in order to construct all of Euclid the center is still required, if provided only a single circular arc in the plane.
Note: In the event that lines are inconveniently parallel, we may resolve the dilemma by reducing the arc. We may choose a point arbitrarily on the arc, e.g. point B`, that truncates it into a smaller arc AB`. In doing this we may ensure that the tangent lines are not parallel, and/or that the lines AB` and m are not parallel.
With this construction, the Poncelet-Steiner theorem can be strengthened, as per Francesco Severi's 1904 theorem.
The center point of a circle is trivially determined from the centerline constructions, which are listed below, and depend on the scenario. The concept for constructing the center point of a circle is fairly straightforward:
How these centerlines are constructed will depend on the information provided, and will be the principle of center constructions in the variety of arrangements to follow.
Two circles without their centers, by itself, is not sufficient information to construct a center. Something more must be given. There are a number of ways to construct the center of a circle given two or more circles, and having additional but sufficient information about the scenario that allows the center to be recovered. This is not an exhaustive list.
Two circles without their centers is not sufficient information to construct a center. Something more must be given, and knowing the fact that they are concentric turns out to be sufficient information. If given two concentric circles devoid of a center, the center may in fact be constructed using a straightedge only.
Given below are two concentric circles (both in green): the inner circle, i, and the outer circle, o.
Two circles without their centers is not sufficient information to construct a center. Something more must be given, and intersection points is sufficient. When two circles are given devoid of their centers, if they intersect we may construct their centers using a Steiner construction.
There are two cases: When the two circles intersect at two points, and alternatively when they intersect at one point (i.e. are tangent to one another). These must be treated separately.
Centers are determined from the intersections of two or more centerlines. To find a circle center in either of the two cases, each of the constructions below must be completed twice, using unique points where arbitrary points are required.
Below two intersecting circles are given (in dark green), devoid of centers, and intersecting at two points U and V.
Below two intersecting circles are given (in dark green), devoid of centers, and intersecting at one point of tangency, the point I.
From any two parallel lines, additional parallels may be constructed. Construct zero-to-four additional parallels, as necessary, such that at least two parallel lines intersect each circle. From two parallel cutting lines through a circle, a centerline through each circle may be constructed perpendicular to these parallel lines. This technique is explained in the centerline from two cutting parallels and circles imply squares sections.
In the general case, there will be two distinct such centerlines, one for each circle. These constructed centerlines are parallel to one another, of which there are two; furthermore, they intersect the original two parallel lines, to which they are perpendicular, thus defining four points of intersection that encompass a rectangle (a parallelogram). From this information it is a triviality to find a second distinct centerline through each circle, using previously described techniques. One may take advantage of the two parallel centerlines to construct a second cutting parallel through each circle, or one may take advantage of the parallelogram to construct parallels of arbitrary lines. Using either approach, two cutting parallels through each circle defines a second distinct centerline.
If from the two original parallel lines only one common centerline through the circles is defined, the construction will fail. It happens by coincidence that the two parallel lines were perpendicular to a common centerline. This is a special case scenario. Constructing the common centerline from two parallels is a similar construction to that from a point on the centerline. An alternative construction exists to find the center points of two circles from the common centerline through them. This construction is explained in the next section.
The common centerline, line c, already counts as a centerline through each circle, and thus one of the two centerlines required to find a center. We need only construct a second centerline through either one or both of the other two circles, thereby allowing a center point to be constructed. The common tangent line, line t, allows us to construct what is needed quite trivially. The center points may be constructed by way of two parallel lines in the plane, as per the previous construction discussed in this section. The parallel lines, however, are themselves trivially constructed.
Let points A, B, C, and D be the four points of intersection between the common centerline, line c, and the two circles. Labeled the points in that same order, such that point A on circle l corresponds to point C on circle r (being the left-most points), and vice versa, point B on circle l corresponds to point D on circle r (being the right-most points). Let also point I be the point of tangency of tangent line t with circle l, and point J be the point of tangency on circle r. It is the case then that lines AI and CJ are parallel, and indeed lines BI and DJ are also parallel. Now, proceed to the previous section to complete the construction.
This construction (and the previous one that is referenced) are used in the following section: constructing a center from two non-intersecting circles having only a centerline in common.
This section is for the treatment of two non-intersecting circles devoid of their centers, but sharing a common centerline, or the scenario involving three non-intersecting circles. These constructions are rather complicated and requires a more thorough discussion. Many hundreds of line constructions are required, and the concepts are advanced, coming from projective geometry and including such topics as inversions, involutions, projections, harmonics, conjugates, homologies, fixed points, and others. Many preliminary constructions must be explained as they will be utilized many times, and they have yet to be discussed in either this subpage, or in the Poncelet-Steiner theorem main article. These are not trivial constructions.
In the two-circle scenario, one centerline common to both circles is already provided as a given to this scenario; to find a circle center we must only find a second centerline to either one of the two circles. In the three-circle scenario, many of the same concepts and constructions will carry over from the two-circle scenario. Refer to my second sandbox to continue the discussion.
Category:Euclidean plane geometry
Category:Theorems in plane geometry
Category:Compass and straightedge constructions
This page associated to the Poncelet–Steiner theorem describes a number of Steiner constructions and related insights and alternative constructions, that are of unique interest for those fascinated by the Poncelet–Steiner theorem and its variants (such as portions of the arc, omission of circle centers, etc.), including the inline references to constructions with parallelograms, squares, etc., as well as other alternative constructions. This page exists to elaborate on the discussions of the Poncelet-Steiner theorem, and develop projective geometry tools to assist in the universality, generality, convenience, and insights of that theorem.
The constructions described below are ordered such that if another construction is required, it is prior listed. No construction requires another listed later, though some may reference constructions found in the Poncelet-Steiner theorem main article. There may also be passive references to other tertiary constructions reserved for yet another subpage. It is assumed that the reader/geometer has read and understood the main article, and has a grasp of the constructions outlined there.
In this section we revisit and generalize two constructions found in the Poncelet-Steiner Theorem main article. The uses for the generalization will be justified in later sections, and the particular cases are demonstrated in this section.
Given a line with segment AB, and a point C (or D) on the line, we may construct the projective harmonic conjugate, point D (or C). The points C and D are each projective harmonic conjugates of one another with respect to, and defined by, the segment AB. Indeed, it is also true that points A and B are each projective harmonic conjugates of one another with respect to the segment CD, though in this case the latter point pair was determined from the former.
The projective harmonic conjugates define a proportionality with the segment AB. It is the case that .
The conjugates may be constructed thus:
Alternatively, point C may be constructed from point D using the same set of lines, in which point Q is found before point X.
The parallel line construction from a bisected line-segment, as seen in the Poncelet-Steiner Theorem article, is a special case of the Projective Harmonic Conjugate construction. In this case, because one of the key defining points of the harmonic is a midpoint, point C, the conjugate point D exists "at infinity" (and thus fails to exist on the line), thereby compelling the line PQ to be parallel. As such, there is no point D on line AB that represents the points of conjugacy. In this construction, however, point D is not the construction of interest, but rather its associated line, the (blue) line PQ.
The construction of the midpoint from two parallels, as seen in the Poncelet-Steiner Theorem article, is a special case of the Projective Harmonic Conjugate construction. In this case, because one of the key defining points, D, of the harmonic does not exist (or rather exists "at infinity"), due to the two lines being parallel, its conjugate placed interior to the segment AB, point C, must be the midpoint. The midpoint construction involves the same set of lines and points as the parallel line construction or the harmonic conjugate construction, simply in a different order and different set of initial conditions.
Given any two lines, m and n, a concurrent line through a point, P, is a line that intersects both lines m and n at the same point that lines m and n intersect one another. In other words, if lines m and n intersect at an unconstructed point Z (off the page, not visible, at great distance), then the concurrent line is the line . Concurrent lines all intersect at the same point; from any two lines, concurrent lines may be constructed through any point in the plane.
The construction is convenient when a third line is required, defined by a point P and the (possibly unconstructed) intersection point between two other lines. This is a particularly useful construction when the point of intersection of the two lines is off the page at a great distance. The circle in the plane is not required.
Given lines m and n (both in black), and a point P through which the concurrent is to be constructed:
It is important to note that whenever point P is equidistant between the two lines m and n (i.e. existing on the angle bisector between them), the concurrent line construction will fail. Alternatively phrased, the construction will fail if the point P is the midpoint of any segment colinear with the transversal line, in which consecutive interior angles are congruent, and the endpoints of the segment exist on the two lines m and n. This is a special case scenario.
A concurrent through P may still be constructed, however. A simple solution is to construct an intermediary concurrent line first.
Given any two lines m and n, and an arbitrary point P in the plane, the concurrent of the lines may be constructed through the point using the projective harmonic conjugate. This construction does not require a circle, and avoids the special case scenario of the projective point method.
Given two parallel lines, m and n, a third parallel through an arbitrary point, P, can be constructed without the use of the circle.
This construction is merely a special case of the concurrent line construction, above, wherein the two given lines are already parallel. Thus, refer to the construction of the concurrent line in the previous section. It will not be repeated here. The parallel nature of lines m and n do not prohibit the concurrent line construction; the same set of constructable lines is achievable.
The principle behind this construction is that parallel lines "intersect at infinity", and concurrent lines intersect at the same point, therefore the concurrent to two parallels is also parallel.
Thus a third parallel through an arbitrary point may be constructed without a circle whenever there are already two parallels. The construction is trivial:
If the point P exists equidistant between the two parallel lines, the same special case is applicable and the construction will fail, if the projective point method is used; use an intermediary parallel. If the harmonic conjugate method is used, this should not be an issue.
Given are three arbitrary lines in space - lines a, b, and c - each parallel to one another, and the line in the middle is equidistant from the other two (i.e. "evenly spaced").
We may find the midpoint of any arbitrary line segment in space, PQ:
In an alternative construction, a midpoint of a line segment may be constructed from a parallelogram. The parallelogram allows for the construction of a line parallel to PQ, upon which the midpoint construction found in the Poncelet-Steiner theorem main article may be utilized.
Whenever three parallels of this type exist in the plane - with one line equidistant between the other two - any arbitrary line cutting through them can have a parallel of it constructed arbitrarily in space, as per steps 1 and 2 in the previous construction, because the three parallels intersect the cutting line such that a bisected segment exists. In the previous construction, points J, L, and S, form three adjacent of the four vertices of a parallelogram, the fourth being unlabeled. Thus, parallelograms are implied, and are arbitrarily constructable anywhere along the three parallels, such that any two of the parallels are opposing sides of the parallelogram. Any constructions involving the parallelogram are available.
Parallelograms are a useful tool in their own right, though not as powerful as a circle.
Given a line m and a parallelogram STUV (both in black), a second parallel can be constructed symmetrically across the parallelogram:
The principle is that parallels induce reflections. The parallelogram is not strictly speaking necessary; what is necessary is that there are two parallels intersecting the line being reflected, and an arbitrary point (X, in this case) that exists equidistant between the two parallels, across which we make reflections. The parallelogram allows for the construction of both the center point and the parallels, and because there are two distinct sets of parallels in a parallelogram we are assured the entire plane is covered.
Alternatively, line segment SU is bisected by point X. Two parallels may be made through points T and V, as per the parallel line constructions found on the Poncelet-Steiner theorem page. These parallels will intersect the given line m, and together with point X a reflection can be made.
Regarding the above construction involving the parallelogram, the parallelogram predetermines the location of the second parallel, as a reflection across the parallelogram. However, once a second parallel is in the plane, a third parallel may be constructed arbitrarily. Whenever a parallelogram exists in the plane, any line may have a parallel constructed from it, through any arbitrary point in the plane.
Two previous constructions are trivially combined:
Given any parallelogram ABCD, three equally spaced parallels may be constructed. Indeed, four distinct sets may be constructed.
Therefore, any constructions based on the existence of three equidistant parallels in the plane are achievable, whenever a parallelogram exists.
Since three equidistant parallels imply a parallelogram, and a parallelogram implies three equidistant parallels, they are equivalent premises in any construction.
Given is a circle, centered at point O, we may construct an arbitrary inscribed parallelogram:
Any constructions involving the parallelogram may be achieved when a circle is provided.
The parallelogram does not imply the circle. The circle (with its center) is a stronger proposition.
Indeed, a circle implies a square, which is stronger than the parallelogram. Any line passing through the center of the circle may have a parallel of it constructed also through the circle. From the intersection points of these lines with the circle, an isosceles trapezoid (or triangle) is defined. The non-parallel sides of the trapezoid may be extended to an intersection point, which along with the circle center defines a perpendicular. The perpendicular intersects the circle at two points. The two perpendicular lines passing through the circles center define the diagonals of a square; the vertices are the points where these lines intersect the circle. Refer to the later construction Constructing a Centerline from Two Cutting Parallels for some details.
The translation of a line segment is a relatively trivial process, and can be done with either a circle or a parallelogram, or by any other means that allows for the construction of parallel lines.
Given is line segment AB (in black), and a point A' to which the endpoint A is to be translated.
This construction will, however, fail if point A' is colinear with line AB. A simple solution for this condition is to translate the line segment AB to an intermediary non-colinear point X first, then translate it back to point A' .
The line segment rotation is another useful transformation, and is of particular use in Steiner constructions, as some constructions will fail under particular scenarios involving the coincidental orientation of line segments. The construction of the radical axis and the intersection between a line and a circle can suffer this scenario.
The Poncelet-Steiner theorem article covers one rotation method, which unfortunately fails when rotating a line segment a full 180 degrees. There is a workaround as explained in the article, but is unnecessarily burdensome. The construction provided below is an alternative approach to line segment rotation that is no more complicated than the one in the main article, and will work for half-circle rotations. This construction, however, has its own fallbacks.
Given is a line segment AB (in black), and an endpoint A about which the rotation is made. This may represent a circle A(B). Provided also is given circle O(r).
As mentioned, if point P is colinear with line AO this construction will fail. We may instead refer to the construction in the Poncelet-Steiner theorem article, or use a sequence of partial rotations to achieve the same ends, as was alternatively done in the main article.
Given is a line segment AB (in black), on and extended by line AB (in red), and two arbitrary parallel lines m and n (both in orange) in the plane. One or both parallels may be constructed if necessary. We may construct on line AB an extension of the line segment to any arbitrary integer multiple. In the image below, segment AC is two times, segment AD is three times, and segment AE is four times the length of segment AB.
Observe that on line m, a sequence of extensions are being generated, corresponding to those on line AB. This second set of segment extensions are scaled down in size, proportional with the distances between the parallel lines. For the purposes of this construction, the proportionality, and thus the parallel line spacing is arbitrary. Specifically, the ratio between segments AB and ab is equal to the ratio of the distance line AB is to line n and the distance line m is to line n. This feature could be intentionally utilized if one has control over the line spacing.
The construction of the secondary segment extension, on line m, is essential for the following section on segment sectioning.
Extensions may also be achieved through a series of colinear translations of the segment, as seen in a previous section, which will require the construction of arbitrary parallels. This construction, though it makes use of parallels, they are not strictly arbitrary and are presumed to already be present.
Given is a segment AB, which we wish to n-sect. In the example below, a trisection is performed. We are also given a parallel line m on which there already exists a trisected segment ad, which was constructed as a three-times extension of segment ab, and as a by product of a three-times extension of segment AB, as seen in the previous construction segment extensions.
Given is a line m (in black) across which reflections are to be made, the point P which we aim to reflect across the line, and the given circle (in green) centered at point O.
If line m passes through circle center O.
To reflect a single point about a line, the following construction is used:
To reflect any figure about a line - be they other lines, circles, or other figures - simply reflect each of its defining points about the line.
The construction is a trivial application of the point reflection:
Given is a square ABDC (in dark green) and an arbitrary line m (in black), and a point in the plane P. We wish to construct a perpendicular of line m through point P.
These constructions concern the construction of tangent lines, points of tangency, pole point, and the polar line, which runs through the points of tangency.
Given is an arbitrary circle (in green) with a center point O (which will not be utilized), and a point P existing on the arc of the circle. We wish to construct the tangent lines to the circle at the point. Such a point has a tangent line that is also the polar.
There are numerous properties involving the polar of which we may take advantage. Several different constructions avail themselves, and are useful under different circumstances.
These constructions work whether point P is internal or external to the circle, but will fail if it is on the arc of the circle or coincident with the circle center. In the former case, refer to the previous construction for a tangent line through a point on the circle. In the latter case, no polar exists.
Given is an arbitrary circle (in green) with a center point O (which will not be utilized), and a point P existing not on the circle. We wish to construct the tangent lines to the circle which pass through the point. Observe the relationships between the points of tangency on the circle, tangent lines through point P, the polar line of point P, and the inverse in the circle of point P.
It is valid to point out that, strictly speaking, points E and F and the line passing through them never needed to be drawn. Points A, B, C and D were sufficient. The point Y may be constructed as the intersection of lines AC and BD, although this is often less convenient. Additionally, point Y need not be constructed; we need only the concurrent between lines AC and BD passing through point X; this is the polar in the circle of point P.
Alternatively, we could have used points A and B (which exist on the same line through point P). From these points we could have constructed tangents to the circle, as per the previous construction for a tangent through a point on the arc of the circle. These tangent lines would intersect one another on the polar at a point X. We can then find point Y, also on the polar, by finding the intersection of the tangents to the circle at points C and D (both of which are colinear with point P). As before, concurrent line constructions may be employed when necessary.
We may also find a point X as the harmonic conjugate of point P on the line AB. Point Y may be found as the harmonic conjugate of point P on the line CD. The projective harmonic conjugate of point P on any cutting line through the circle exists on the polar. It should be noted that any chord through the circle on the line passing through point P defines a line segment upon which the projective harmonic conjugate of the point lies on the polar. Using multiple such chords, multiple points on the polar may be constructed, thus defining the line.
This is a relatively trivial construction. Let point P exist in the plane other than the center of the circle, point O. For point O no inverse exists.
For any circle with radius r and center point O, any point P in the plane and its inverse (point I) in the circle share a proportionality relationship. It is the case that .
Let also a line exist through point P that is centerline through the circle (that is, it passes through the center of the circle). The center point of the circle need not be constructed or provided. Naturally if the centerline does not exist, it must be constructed, either directly as the line through the circle center or through some other means.
The inverse of point P may be constructed simply.
If point P exists on the arc of the circle, nothing need be done. The point is its own inverse.
Let point P exist in the plane internal or external to the circle:
Alternatively, if the point P is internal the circle, a line through it perpendicular to the centerline may be constructed. Where this perpendicular intersects the circle, tangents to the circle may be constructed, which intersect one another at the inverse point. The line perpendicular to the centerline through point P may be constructed as the third parallel to the two tangents to the circle at the points the centerline intersect the circle.
Given the polar construction of the previous section using the projective harmonic conjugate technique, if the chord is chosen to be the diameter of the circle, the constructed conjugate is also the inverse of a point P in a circle.
Given an arbitrary line m in space and a circle (devoid of its center), we may find the point M that is the pole to the polar m in the circle.
This construction will work for polar lines that are both intersecting and non-intersecting to the circle.
This construction is fully general. Each of the special cases to follow are simplified special cases of this construction.
This is a trivial case. If the polar is tangential to the circle, the point of intersection to the circle is the pole. The pole is on the polar at the point of tangency. There is nothing to construct, but to find the point of tangency by intersecting the circle with the tangent.
For any polar intersecting the circle in two points A and B, we may construct the tangents to the circles at the points A and B. These tangent lines intersect one another at a unique point exterior to the circle. The intersection point is the pole.
If the polar is also a centerline (thus crossing the circle center) then there is no pole; or it "exists at infinity". The tangent lines to the circle, in this case, will be parallel.
Given are two parallel lines and a circle placed arbitrarily in the plane. Construct additional parallel lines as necessary, such that no fewer than two intersect the circle, with a total of four distinct intersection points between them.
Let one of the parallel lines intersect the circle at points A and B; let a second parallel intersect at points C and D. A centerline to the circle may be constructed simply:
Observe that a single midpoint construction (or segment bisection) may be used. Line segments AB and CD may be bisected concurrently, and bisected by the very centerline we wish to construct. The details may be found in the appropriate section on the Poncelet-Steiner theorem main article.
More concisely, we may define points X and Y by the intersections of lines AC with BD and AD with BC, respectively. The line XY is the desired centerline (which bisects both AB and CD concurrently).
Should a point (e.g. X) not exist due to two lines (e.g. AC with BD) being parallel, we may use concurrent line constructions, or we may use the two parallel lines AB and CD to construct a third parallel through the circle at a more convenient location for this construction. Alternatively, we may recognize that this scenario can only happen if ABCD is a rectangle, in which case each diagonal defines a centerline.
If one of the parallels happens to pass through the circle center, its intersection points with the circle defines two opposing vertices of a square inscribed in the circle. The constructed centerline, which is orthogonal and also passes through the circle center, defines the other two vertices. Thus a square may be defined, as per a previous section Circles Imply Squares.
Alternatively, we may find the two distinct poles corresponding to the two distinct parallel lines (treated as polars) to the circle. The two poles of these parallels define a perpendicular line which passes through the circle center.
Given two non-intersecting circles (both in dark green), each devoid of their centers, and given an arbitrary point on the centerline, point M, exterior to at least one of the two circles, it is possible to construct the centerline.
If the point M is in the interior of one of the circles, only a slight modification is necessary. The polar through only one of the circles is truly necessary. Tangent lines through the point M to any circle they can be constructed on will intersect the other circle at two points for each tangent line; they will be symmetric across the centerline. In the general case at least two additional centerline points may be constructed as intersections.
This construction will fail if the point M is interior to both circles simultaneously, one interior to the other but not intersecting. It will also fail if the point is the common point of tangency between the two circles, be they interior or exterior.
In general, from any point on the centerline, two parallel lines may be constructed which are perpendicular to the centerline. From this scenario, refer back to the centerline construction from two parallel lines in the plane.
When two arbitrary non-intersecting circles are provided - left and right circles l and r - without their centers, and a point P exists in the plain on the radical axis between the circles, the common centerline through the circles may be constructed.
This is a relatively trivial construction. The intersection point of the polars to each circle from the point on the radical axis, is also on the radical axis, and thus define the line:
Given is an arc of a circle (in dark green), of arbitrary arc length, spanning between the endpoints A and B of the arc AB. Given also is a line m (in black). We wish to intersect the line with the circle of the arc; that is to say, we wish to find the intersection points of the line with the circle which is defined by the arc.
Notice that for this construction the center, point O, of said circle is not utilized. It need not be provided at all in order to intersect a line with the arc of the circle. However, according to Steiner's theorem, in order to construct all of Euclid the center is still required, if provided only a single circular arc in the plane.
Note: In the event that lines are inconveniently parallel, we may resolve the dilemma by reducing the arc. We may choose a point arbitrarily on the arc, e.g. point B`, that truncates it into a smaller arc AB`. In doing this we may ensure that the tangent lines are not parallel, and/or that the lines AB` and m are not parallel.
With this construction, the Poncelet-Steiner theorem can be strengthened, as per Francesco Severi's 1904 theorem.
The center point of a circle is trivially determined from the centerline constructions, which are listed below, and depend on the scenario. The concept for constructing the center point of a circle is fairly straightforward:
How these centerlines are constructed will depend on the information provided, and will be the principle of center constructions in the variety of arrangements to follow.
Two circles without their centers, by itself, is not sufficient information to construct a center. Something more must be given. There are a number of ways to construct the center of a circle given two or more circles, and having additional but sufficient information about the scenario that allows the center to be recovered. This is not an exhaustive list.
Two circles without their centers is not sufficient information to construct a center. Something more must be given, and knowing the fact that they are concentric turns out to be sufficient information. If given two concentric circles devoid of a center, the center may in fact be constructed using a straightedge only.
Given below are two concentric circles (both in green): the inner circle, i, and the outer circle, o.
Two circles without their centers is not sufficient information to construct a center. Something more must be given, and intersection points is sufficient. When two circles are given devoid of their centers, if they intersect we may construct their centers using a Steiner construction.
There are two cases: When the two circles intersect at two points, and alternatively when they intersect at one point (i.e. are tangent to one another). These must be treated separately.
Centers are determined from the intersections of two or more centerlines. To find a circle center in either of the two cases, each of the constructions below must be completed twice, using unique points where arbitrary points are required.
Below two intersecting circles are given (in dark green), devoid of centers, and intersecting at two points U and V.
Below two intersecting circles are given (in dark green), devoid of centers, and intersecting at one point of tangency, the point I.
From any two parallel lines, additional parallels may be constructed. Construct zero-to-four additional parallels, as necessary, such that at least two parallel lines intersect each circle. From two parallel cutting lines through a circle, a centerline through each circle may be constructed perpendicular to these parallel lines. This technique is explained in the centerline from two cutting parallels and circles imply squares sections.
In the general case, there will be two distinct such centerlines, one for each circle. These constructed centerlines are parallel to one another, of which there are two; furthermore, they intersect the original two parallel lines, to which they are perpendicular, thus defining four points of intersection that encompass a rectangle (a parallelogram). From this information it is a triviality to find a second distinct centerline through each circle, using previously described techniques. One may take advantage of the two parallel centerlines to construct a second cutting parallel through each circle, or one may take advantage of the parallelogram to construct parallels of arbitrary lines. Using either approach, two cutting parallels through each circle defines a second distinct centerline.
If from the two original parallel lines only one common centerline through the circles is defined, the construction will fail. It happens by coincidence that the two parallel lines were perpendicular to a common centerline. This is a special case scenario. Constructing the common centerline from two parallels is a similar construction to that from a point on the centerline. An alternative construction exists to find the center points of two circles from the common centerline through them. This construction is explained in the next section.
The common centerline, line c, already counts as a centerline through each circle, and thus one of the two centerlines required to find a center. We need only construct a second centerline through either one or both of the other two circles, thereby allowing a center point to be constructed. The common tangent line, line t, allows us to construct what is needed quite trivially. The center points may be constructed by way of two parallel lines in the plane, as per the previous construction discussed in this section. The parallel lines, however, are themselves trivially constructed.
Let points A, B, C, and D be the four points of intersection between the common centerline, line c, and the two circles. Labeled the points in that same order, such that point A on circle l corresponds to point C on circle r (being the left-most points), and vice versa, point B on circle l corresponds to point D on circle r (being the right-most points). Let also point I be the point of tangency of tangent line t with circle l, and point J be the point of tangency on circle r. It is the case then that lines AI and CJ are parallel, and indeed lines BI and DJ are also parallel. Now, proceed to the previous section to complete the construction.
This construction (and the previous one that is referenced) are used in the following section: constructing a center from two non-intersecting circles having only a centerline in common.
This section is for the treatment of two non-intersecting circles devoid of their centers, but sharing a common centerline, or the scenario involving three non-intersecting circles. These constructions are rather complicated and requires a more thorough discussion. Many hundreds of line constructions are required, and the concepts are advanced, coming from projective geometry and including such topics as inversions, involutions, projections, harmonics, conjugates, homologies, fixed points, and others. Many preliminary constructions must be explained as they will be utilized many times, and they have yet to be discussed in either this subpage, or in the Poncelet-Steiner theorem main article. These are not trivial constructions.
In the two-circle scenario, one centerline common to both circles is already provided as a given to this scenario; to find a circle center we must only find a second centerline to either one of the two circles. In the three-circle scenario, many of the same concepts and constructions will carry over from the two-circle scenario. Refer to my second sandbox to continue the discussion.
Category:Euclidean plane geometry
Category:Theorems in plane geometry
Category:Compass and straightedge constructions