Mock Geometry AIME 2011 Problems/Problem 3
Problem
In triangle 
Points 
and 
are located on 
such that 
The circumcircle of 
cuts 
at 
respectively. If 
then the ratio 
can be expressed in the form 
where 
are relatively prime positive integers. Find 
Solution
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]](http://latex.artofproblemsolving.com/0/7/d/07d435eb6a55b041b35df03b02b9cb0cd4e7571d.png)
By the Power of a Point Theorem on 
, we have 
. By the Power of a Point on 
, we have 
. Dividing these two results yields 
. We are given 
and so 
. Then the previous equation simplifies to 
. Hence 
