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Geometry that "looks" hard

Source: Iran 2nd round 2022 P6

May 9, 2022

geometry

Problem Statement

we have an isogonal triangle

ABC

such that

BC=AB

. take a random

P

on the altitude from

B

to

AC

. The circle

(ABP)

intersects

AC

second time in

M

. Take

(APB)

such that it's on the segment

AC

and

AM=NC

and

M \neq N

.The second intersection of

NP

and circle

(APB)

is

X

, (

X \neq P

) and the second intersection of

AB

and circle

(APN)

is

Y

,(

Y \neq A

).The tangent from

A

to the circle

(APN)

intersects the altitude from

B

at

Z

. Prove that

CZ

is tangent to circle

(PXY)

.

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