Difference between revisions of "2019 IMO Problems/Problem 6"

(Created page with "==Problem== Let I be the incentre of acute triangle ABC with AB ̸= AC. The incircle ω of ABC is tangent to sides BC, CA, and AB at D, E, and F, respectively. The line throug...")
 
m (Problem)
Line 1: Line 1:
 
==Problem==
 
==Problem==
Let I be the incentre of acute triangle ABC with AB ̸= AC. The incircle ω of ABC is tangent to sides BC, CA, and AB at D, E, and F, respectively. The line through D perpendicular to EF meets ω again at R. Line AR meets ω again at P. The circumcircles of triangles PCE and PBF meet again at Q.
+
Let <math>I</math> be the incenter of acute triangle <math>ABC</math> with <math>AB \neq AC</math>. The incircle ω of <math>ABC</math> is tangent to sides <math>BC</math>, <math>CA</math>, and <math>AB</math> at <math>D</math>, <math>E</math>, and <math>F</math>, respectively. The line through <math>D</math> perpendicular to <math>EF</math> meets ω again at <math>R</math>. Line <math>AR</math> meets ω again at P. The circumcircles of triangles PCE and PBF meet again at Q.
Prove that lines DI and PQ meet on the line through A perpendicular to AI.
+
Prove that lines <math>DI</math> and <math>PQ</math> meet on the line through <math>A</math> perpendicular to <math>AI</math>.

Revision as of 22:31, 26 May 2020

Problem

Let $I$ be the incenter of acute triangle $ABC$ with $AB \neq AC$. The incircle ω of $ABC$ is tangent to sides $BC$, $CA$, and $AB$ at $D$, $E$, and $F$, respectively. The line through $D$ perpendicular to $EF$ meets ω again at $R$. Line $AR$ meets ω again at P. The circumcircles of triangles PCE and PBF meet again at Q. Prove that lines $DI$ and $PQ$ meet on the line through $A$ perpendicular to $AI$.

Invalid username
Login to AoPS