Track / Stage 4 / 193 of 340 #453 of 2000

Problem 453

AMC 12 late, AIME early

Geometry Difficulty 4.8 Prove it Berkeley Math Circle: Monthly Contest 6 · United States

Problem:

Let $ABC$ be a triangle, and let $X, Y, Z$ be the excenters opposite $A, B, C$ . The incircle of triangle $ABC$ touches $BC, CA, AB$ at points $D, E, F$ . Finally, let $I$ and $O$ denote the incenter and circumcenter of triangle $ABC$ .
Prove that lines $DX, EY, FZ, IO$ are concurrent.

This one wants a proof. Work it on paper, read the official solution, then mark yourself honestly — the ladder only means something if the record is true.

Official solution

Solution:

The fact that $DX, EY, FZ$ are concurrent follows from the fact that triangles $DEF$ and $XYZ$ are homothetic; indeed, note that $EF$ and $YZ$ are both perpendicular to the internal angle bisector of $\angle BAC$ .

Now, to see that the concurrence point lies on $IO$ , note that point $I$ is the orthocenter of triangle $XYZ$ , and $O$ is the nine-point center of triangle $XYZ$ . Thus line $IO$ is the Euler line of triangle $XYZ$ and thus passes through the circumcenter $S$ of triangle $XYZ$ . But $I$ is the circumcenter of triangle $DEF$ , hence line $SI$ passes through the concurrency point.

Source: MathNet, licensed CC-BY-4.0. Statement and solution reproduced as published; topic, difficulty and ordering added by this site.