MathDB

Hard excircle geo

Source: MEMO 2023 T6

August 25, 2023

geometry

Problem Statement

Let

ABC

be an acute triangle with

AB < AC

. Let

J

be the center of the

A

-excircle of

ABC

. Let

D

be the projection of

J

on line

BC

. The internal bisectors of angles

BDJ

and

JDC

intersectlines

BJ

and

A

at

X

and

Y

, respectively. Segments

XY

and

JD

intersect at

P

. Let

Q

be the projection of

A

on line

BC

. Prove that the internal angle bisector of

QAP

is perpendicular to line

XY

.
Proposed by Dominik Burek, Poland

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