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Reflection of incentre on a side

Source: Indian Team Selection Test 2015 Day 3 Problem 1

July 11, 2015

geometryreflectionincenter

Problem Statement

In a triangle

ABC

, a point

D

is on the segment

BC

, Let

X

and

Y

be the incentres of triangles

ACD

and

ABD

respectively. The lines

BY

and

CX

intersect the circumcircle of triangle

AXY

at

P\ne Y

and

Q\ne X

, respectively. Let

K

be the point of intersection of lines

PX

and

QY

. Suppose

K

is also the reflection of

I

in

BC

where

I

is the incentre of triangle

ABC

. Prove that

\angle BAC=\angle ADC=90^{\circ}

.

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