Let H be the point of intersection of the diagonal AD and BG of heptagon ABCDEFG. Let r be the perpendicular distance of point H to side AB.
Consider a scalene triangle whose vertices are some suitably chosen three vertices of the same regular heptagon. Let its inradius be R.
What is the ratio R : r?
The two lengths are the same. R:r = 1:1
The scalene triangle can only be formed with two adjacent vertices and a third that is 2 vertices away from one (4 away from the other.) There are 14 such triangles, all congruent.
Choose as the scalene triangle ABF. Its center is the intersection of the angle bisectors, two of which are AD and BG. These intersect, of course, at H. The radius R is the perpendicular distance from H to a side. One of the sides is AB, so R=r.

Posted by Jer
on 20181204 11:47:27 