Grigory MerzonTriangulate a cyclic polygon. “Japanese theorem”: the sum of inradii of triangles doesn't depend on the triangulation.
Moreover, this sum is constant in the “Poncelet family” of all polygons with the same incircle and circumcircle.
Dave Richeson@g_merzon I love the "Japanese theorem"! I spent a while thinking about it over a decade ago. I wrote something for the MAA's online journal LOCI. Here it is in case you haven't seen it and you think it is useful to your investigations: https://www.maa.org/press/periodicals/the-japanese-theorem-for-nonconvex-polygons-a-japanese-temple-problem
@divbyzero Thank you! In fact I was reading https://www.maa.org/press/periodicals/loci/the-japanese-theorem-for-nonconvex-polygons-the-total-inradius-function when I've received your reply )
@g_merzon Totally unrelated to the content, I happen to remember that as one of the most frustrating publication experiences of my career. I submitted it years earlier, it was accepted, and then I was ghosted by the editor for years. I ended up reaching out to others at the MAA, and eventually it was posted online. By the time it came out, I didn't want to think about it anymore! 😂 (I think I found out afterward that the editor was struggling with personal issues at that time.)
@divbyzero yes, very frustrating : (