MathDB

Inversely Similiar Triangles

Source: JMO/5 2016

April 20, 2016

geometrysimilar triangles2016 USAJMOUSAJMOUSA(J)MOisogonal conjugatesAngle Chasing

Problem Statement

Let

\triangle ABC

be an acute triangle, with

O

as its circumcenter. Point

H

is the foot of the perpendicular from

A

to line

\overleftrightarrow{BC}

, and points

P

and

Q

are the feet of the perpendiculars from

H

to the lines

\overleftrightarrow{AB}

and

\overleftrightarrow{AC}

, respectively.
Given that

AH^2=2\cdot AO^2,

prove that the points

O,P,

and

Q

are collinear.

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