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2019 IGO Advanced P1

Source: 6th Iranian Geometry Olympiad (Advanced) P1

September 20, 2019

IGOIrangeometryAngle Chasing

Problem Statement

Circles

\omega_1

and

\omega_2

intersect each other at points

A

and

B

. Point

C

lies on the tangent line from

A

to

\omega_1

such that

\angle ABC = 90^\circ

. Arbitrary line

\ell

passes through

C

and cuts

\omega_2

at points

P

and

Q

. Lines

AP

and

AQ

cut

\omega_1

for the second time at points

X

and

Z

respectively. Let

Y

be the foot of altitude from

A

to

\ell

. Prove that points

X, Y

and

Z

are collinear.
Proposed by Iman Maghsoudi

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