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Incircle and excircle of triangles of a cyclic quadrilateral

Source: 2012 Indonesia Round 2 TST 3 Problem 3

March 18, 2012

geometrycircumcircleincentercyclic quadrilateralprojective geometrygeometry unsolved

Problem Statement

Given a cyclic quadrilateral

ABCD

with the circumcenter

O

, with

BC

and

AD

not parallel. Let

P

be the intersection of

AC

and

BD

. Let

E

be the intersection of the rays

AB

and

DC

. Let

I

be the incenter of

EBC

and the incircle of

EBC

touches

BC

at

AD

. Let

J

be the excenter of

EAD

that touches

AD

and the excircle of

EAD

that touches

AD

touches

AD

at

T_2

. Let

Q

be the intersection between

IT_1

and

JT_2

. Prove that

O,P,Q

are collinear.

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