IGRP Episode 17 Seminar 8. Problem 18. Chain Of Pappus

Описание: In this seminar we solve a very illustrative problem, known as Chain Of Pappus, that brings front and center the tactical might of the Transform-And-Conquer problem-solving approach.

As far as we know, Pappus of Alexandria was possibly aware of inversion of the plane in a circle but he managed to solve the problem that is now named after him using nothing but what we now call "elementary plane Euclidean geometry". The solution of Pappus runs for several pages and is not easy on the eye.

In contrast, a fruitful inversion of the parent plane turns this elementary but very difficult problem into a walk in the park, a problem that can be solved nowadays by any sufficiently motivated middle school student.

A key observation in this problem is based on the theorem that we proved earlier in the course: any circle that passes through the center of inversion is taken into a straight line.

By making the point of tangency, T, of the two generating circles the center inversion, we reflect these two circles into two straight lines that are parallel, which implies that the complicated family of the original inscribed circles is taken by that transformation into a plain vanilla family of circles of equal radii!

Once that maneuver is understood, what follows next is a run-of-the-mill torture of two pairs of advantageous right triangles and we are home free.