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Incenters and concyclic points

Source: China Mathematical Olympiad 2018 Q4

November 16, 2017

geometry

Problem Statement

ABCD

is a cyclic quadrilateral whose diagonals intersect at

P

. The circumcircle of

\triangle APD

meets segment

AB

at points

A

and

E

. The circumcircle of

\triangle BPC

meets segment

AB

at points

B

and

F

. Let

I

and

J

be the incenters of

\triangle ADE

and

\triangle BCF

, respectively. Segments

IJ

and

AC

meet at

K

. Prove that the points

A,I,K,E

are cyclic.

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