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Two tangent circles

Source: 2020 Korean MO winter camp Test 1 P5

September 7, 2020

geometry

Problem Statement

\square ABCD

is a quadrilateral with

\angle A=2\angle C <90^\circ

.

I

is the incenter of

\triangle BAD

, and the line passing

I

and perpendicular to

AI

meets rays

CB

and

CD

at

E,F

respectively. Denote

O

as the circumcenter of

\triangle CEF

. The line passing

E

and perpendicular to

OE

meets ray

OF

at

Q

, and the line passing

F

and perpendicular to

OF

meets ray

OE

at

P

. Prove that the circle with diameter

PQ

is tangent to the circumcircle of

\triangle BCD

.

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