# Mock Geometry AIME 2011 Problems/Problem 3

## Problem

In triangle $ABC,$ $BC=9.$ Points $P$ and $Q$ are located on $BC$ such that $BP=PQ=2,$ $QC=5.$ The circumcircle of $APQ$ cuts $AB,AC$ at $D,E$ respectively. If $BD=CE,$ then the ratio $\frac{AB}{AC}$ can be expressed in the form $\frac{m}{n},$ where $m,n$ are relatively prime positive integers. Find $m+n.$

## Solution

$[asy] unitsize(1cm); draw((-1,0)--(8,0)--(0,4)--cycle); draw((0,4)--(1,0)); draw((0,4)--(3,0)); label("A",(0,4),N); label("B",(-1,0),WSW); label("C",(8,0),ESE); label("P",(1,0),S); label("Q",(3,0),S); draw(circumcircle((0,4),(1,0),(3,0))); label("D",(-0.6,2),SW); label("E",(4.5,1.7),NE); [/asy]$

By the Power of a Point Theorem on $B$, we have $BD*BA=BP \times BQ=2 \times 4=8$. By the Power of a Point on $C$, we have $CE \times CA=CQ \times CP=5 \times 7=35$. Dividing these two results yields $\frac{BD \times BA}{CE \times CA}=\frac{8}{35}$. We are given $BD=CE$ and so $\frac{BD}{CE}=1$. Then the previous equation simplifies to $\frac{AB}{AC}=\frac{8}{35}$. Hence $m+n=8+35=\boxed{043}$