Difference between revisions of "Mock Geometry AIME 2011 Problems/Problem 3"
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==Problem== | ==Problem== | ||
In triangle <math>ABC,</math> <math>BC=9.</math> Points <math>P</math> and <math>Q</math> are located on <math>BC</math> such that <math>BP=PQ=2,</math> <math>QC=5.</math> The circumcircle of <math>APQ</math> cuts <math>AB,AC</math> at <math>D,E</math> respectively. If <math>BD=CE,</math> then the ratio <math>\frac{AB}{AC}</math> can be expressed in the form <math>\frac{m}{n},</math> where <math>m,n</math> are relatively prime positive integers. Find <math>m+n.</math> | In triangle <math>ABC,</math> <math>BC=9.</math> Points <math>P</math> and <math>Q</math> are located on <math>BC</math> such that <math>BP=PQ=2,</math> <math>QC=5.</math> The circumcircle of <math>APQ</math> cuts <math>AB,AC</math> at <math>D,E</math> respectively. If <math>BD=CE,</math> then the ratio <math>\frac{AB}{AC}</math> can be expressed in the form <math>\frac{m}{n},</math> where <math>m,n</math> are relatively prime positive integers. Find <math>m+n.</math> | ||
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==Solution== | ==Solution== |
Revision as of 21:05, 1 January 2012
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
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