Track / Stage 4 / 317 of 340 #577 of 1964

Problem 577

AMC 12 late, AIME early

Geometry Difficulty 5.0 Find the answer HMMT_2

Let $A B C$ be a triangle with $A B=13, B C=14, C A=15$ . Let $O$ be the circumcenter of $A B C$ . Find the distance between the circumcenters of triangles $A O B$ and $A O C$ .

Your answer

A number or a short expression. Spacing, $ signs and \frac vs / are all fine.

Official solution

Let $S, T$ be the intersections of the tangents to the circumcircle of $A B C$ at $A, C$ and at $A, B$ respectively. Note that $A S C O$ is cyclic with diameter $S O$ , so the circumcenter of $A O C$ is the midpoint of $O S$ , and similarly for the other side. So the length we want is $\frac{1}{2} S T$ . The circumradius $R$ of $A B C$ can be computed by Heron's formula and $K=\frac{a b c}{4 R}$ , giving $R=\frac{65}{8}$ . A few applications of the Pythagorean theorem and similar triangles gives $A T=\frac{65}{6}, A S=\frac{39}{2}$ , so the answer is $\frac{91}{6}$

Source: Omni-MATH, licensed Apache-2.0. Statement and solution reproduced as published; topic, difficulty and ordering added by this site.