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

(→Problem) |
(→Problem) |
||

Line 1: | Line 1: | ||

==Problem== | ==Problem== | ||

− | Let <math>I</math> be the incenter of acute triangle <math>ABC</math> with <math>AB \neq AC</math>. The incircle | + | Let <math>I</math> be the incenter of acute triangle <math>ABC</math> with <math>AB \neq AC</math>. The incircle <math>\omega</math> 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 <math>P</math>. The circumcircles of triangles <math>PCE</math> and <math>PBF</math> meet again at <math>Q</math>. |

Prove that lines <math>DI</math> and <math>PQ</math> meet on the line through <math>A</math> perpendicular to <math>AI</math>. | Prove that lines <math>DI</math> and <math>PQ</math> meet on the line through <math>A</math> perpendicular to <math>AI</math>. |

## Latest revision as of 16:04, 25 June 2020

## Problem

Let be the incenter of acute triangle with . The incircle of is tangent to sides , , and at , , and , respectively. The line through perpendicular to meets ω again at . Line meets ω again at . The circumcircles of triangles and meet again at . Prove that lines and meet on the line through perpendicular to .