2016 APMO Problems/Problem 1: Difference between revisions
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==Problem== | ==Problem== | ||
We say that a triangle <math>ABC</math> is great if the following holds: for any point <math>D</math> on the side <math>BC</math>, if <math>P</math> and <math>Q</math> are the feet of the perpendiculars from <math>D</math> to the lines <math>AB</math> and <math>AC</math>, respectively, then the reflection of <math>D</math> in the line <math>PQ</math> lies on the circumcircle of the triangle <math>ABC</math>. Prove that triangle <math>ABC</math> is great if and only if <math>\angle A = 90^{\circ}</math> and <math>AB = AC</math>. | We say that a triangle <math>ABC</math> is great if [[the]] following holds: for any point <math>D</math> on the side <math>BC</math>, if <math>P</math> and <math>Q</math> are the feet of the perpendiculars from <math>D</math> to the lines <math>AB</math> and <math>AC</math>, respectively, then the reflection of <math>D</math> in the line <math>PQ</math> lies on the circumcircle of the triangle <math>ABC</math>. Prove that triangle <math>ABC</math> is great if and only if <math>\angle A = 90^{\circ}</math> and <math>AB = AC</math>. | ||
==Solution== | ==Solution== | ||
Latest revision as of 15:52, 25 March 2024
Problem
We say that a triangle 
is great if the following holds: for any point 
on the side 
, if 
and 
are the feet of the perpendiculars from to the lines 
and 
, respectively, then the reflection of in the line 
lies on the circumcircle of the triangle . Prove that triangle is great if and only if 
and 
.
