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By the n=3 case of Poncelet's closure theorem for circles, if the equation of Euler's theorem in geometry is satisfied, relating the inradius, circumradius, and distance between the incenter and circumcenter, then for given incircle and circumcircle, every point A on the circumcircle is a vertex of a triangle ABC having those circles. As we walk vertex A completely around the circumcircle from a given starting point X, dragging B and C along (with slippage) so as to maintain the given incircle, the original triangle occurs a total of three times: when A=X, when B=X, and when C=X.
For each of the three portions of this walkaround, what is the nature of the closed path traced out by (1) the triangle's centroid, and (2) its orthocenter? Does it trace a locus of a particular named type? Does the locus enclose a convex set? Loraof ( talk) 01:22, 16 April 2017 (UTC)
The question can be visualized by this figure from the article, in which unfortunately the animation is too fast. I'm interested especially in the locus of centroids. Loraof ( talk) 14:21, 17 April 2017 (UTC)
Thanks to both of you! Loraof ( talk) 14:42, 18 April 2017 (UTC)
How many binary ratios are needed for an equivalent representation of a ternary ratio of numbers a:b:c? What is the general case for n-term ratios a:b:c:d:.:....:q? (Thanks.)-- 82.137.11.181 ( talk) 13:47, 16 April 2017 (UTC)
For number of dimensions n, take the n-hypercube of edge 2 units centered at the origin. Circumscribe the n-hypersphere around it. Divide the hypersphere's volume into n parts (Vn,V(n-1), ... V0) based on how many of coordinates of each point have absolute value<1. Is Vn (the n-hypercube) always the largest volume of the parts for each number of dimensions? For example if n=4, does the hypercube have more hypervolume than the points where only 3 of the 4 coordinates have absolute value<1? Naraht ( talk) 18:03, 16 April 2017 (UTC)
Mathematics desk | ||
---|---|---|
< April 15 | << Mar | April | May >> | Current desk > |
Welcome to the Wikipedia Mathematics Reference Desk Archives |
---|
The page you are currently viewing is a transcluded archive page. While you can leave answers for any questions shown below, please ask new questions on one of the current reference desk pages. |
By the n=3 case of Poncelet's closure theorem for circles, if the equation of Euler's theorem in geometry is satisfied, relating the inradius, circumradius, and distance between the incenter and circumcenter, then for given incircle and circumcircle, every point A on the circumcircle is a vertex of a triangle ABC having those circles. As we walk vertex A completely around the circumcircle from a given starting point X, dragging B and C along (with slippage) so as to maintain the given incircle, the original triangle occurs a total of three times: when A=X, when B=X, and when C=X.
For each of the three portions of this walkaround, what is the nature of the closed path traced out by (1) the triangle's centroid, and (2) its orthocenter? Does it trace a locus of a particular named type? Does the locus enclose a convex set? Loraof ( talk) 01:22, 16 April 2017 (UTC)
The question can be visualized by this figure from the article, in which unfortunately the animation is too fast. I'm interested especially in the locus of centroids. Loraof ( talk) 14:21, 17 April 2017 (UTC)
Thanks to both of you! Loraof ( talk) 14:42, 18 April 2017 (UTC)
How many binary ratios are needed for an equivalent representation of a ternary ratio of numbers a:b:c? What is the general case for n-term ratios a:b:c:d:.:....:q? (Thanks.)-- 82.137.11.181 ( talk) 13:47, 16 April 2017 (UTC)
For number of dimensions n, take the n-hypercube of edge 2 units centered at the origin. Circumscribe the n-hypersphere around it. Divide the hypersphere's volume into n parts (Vn,V(n-1), ... V0) based on how many of coordinates of each point have absolute value<1. Is Vn (the n-hypercube) always the largest volume of the parts for each number of dimensions? For example if n=4, does the hypercube have more hypervolume than the points where only 3 of the 4 coordinates have absolute value<1? Naraht ( talk) 18:03, 16 April 2017 (UTC)