In plane geometry, a mixtilinear incircle of a triangle is a circle which is tangent to two of its sides and internally tangent to its circumcircle. The mixtilinear incircle of a triangle tangent to the two sides containing vertex is called the -mixtilinear incircle. Every triangle has three unique mixtilinear incircles, one corresponding to each vertex.
The - excircle of triangle is unique. Let be a transformation defined by the composition of an inversion centered at with radius and a reflection with respect to the angle bisector on . Since inversion and reflection are bijective and preserve touching points, then does as well. Then, the image of the - excircle under is a circle internally tangent to sides and the circumcircle of , that is, the -mixtilinear incircle. Therefore, the -mixtilinear incircle exists and is unique, and a similar argument can prove the same for the mixtilinear incircles corresponding to and . [1]
The -mixtilinear incircle can be constructed with the following sequence of steps. [2]
This construction is possible because of the following fact:
The incenter is the midpoint of the touching points of the mixtilinear incircle with the two sides.
Let be the circumcircle of triangle and be the tangency point of the -mixtilinear incircle and . Let be the intersection of line with and be the intersection of line with . Homothety with center on between and implies that are the midpoints of arcs and respectively. The inscribed angle theorem implies that and are triples of collinear points. Pascal's theorem on hexagon inscribed in implies that are collinear. Since the angles and are equal, it follows that is the midpoint of segment . [1]
The following formula relates the radius of the incircle and the radius of the -mixtilinear incircle of a triangle :
where is the magnitude of the angle at .
[3]
and are cyclic quadrilaterals. [4]
is the center of a spiral similarity that maps to respectively. [1]
The three lines joining a vertex to the point of contact of the circumcircle with the corresponding mixtilinear incircle meet at the external center of similitude of the incircle and circumcircle. [3] The Online Encyclopedia of Triangle Centers lists this point as X(56). [6] It is defined by trilinear coordinates: and barycentric coordinates:
The radical center of the three mixtilinear incircles is the point which divides in the ratio: where are the incenter, inradius, circumcenter and circumradius respectively. [5]
In plane geometry, a mixtilinear incircle of a triangle is a circle which is tangent to two of its sides and internally tangent to its circumcircle. The mixtilinear incircle of a triangle tangent to the two sides containing vertex is called the -mixtilinear incircle. Every triangle has three unique mixtilinear incircles, one corresponding to each vertex.
The - excircle of triangle is unique. Let be a transformation defined by the composition of an inversion centered at with radius and a reflection with respect to the angle bisector on . Since inversion and reflection are bijective and preserve touching points, then does as well. Then, the image of the - excircle under is a circle internally tangent to sides and the circumcircle of , that is, the -mixtilinear incircle. Therefore, the -mixtilinear incircle exists and is unique, and a similar argument can prove the same for the mixtilinear incircles corresponding to and . [1]
The -mixtilinear incircle can be constructed with the following sequence of steps. [2]
This construction is possible because of the following fact:
The incenter is the midpoint of the touching points of the mixtilinear incircle with the two sides.
Let be the circumcircle of triangle and be the tangency point of the -mixtilinear incircle and . Let be the intersection of line with and be the intersection of line with . Homothety with center on between and implies that are the midpoints of arcs and respectively. The inscribed angle theorem implies that and are triples of collinear points. Pascal's theorem on hexagon inscribed in implies that are collinear. Since the angles and are equal, it follows that is the midpoint of segment . [1]
The following formula relates the radius of the incircle and the radius of the -mixtilinear incircle of a triangle :
where is the magnitude of the angle at .
[3]
and are cyclic quadrilaterals. [4]
is the center of a spiral similarity that maps to respectively. [1]
The three lines joining a vertex to the point of contact of the circumcircle with the corresponding mixtilinear incircle meet at the external center of similitude of the incircle and circumcircle. [3] The Online Encyclopedia of Triangle Centers lists this point as X(56). [6] It is defined by trilinear coordinates: and barycentric coordinates:
The radical center of the three mixtilinear incircles is the point which divides in the ratio: where are the incenter, inradius, circumcenter and circumradius respectively. [5]