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A weird conditional geometry

Source: 2023 Taiwan Mathematics Olympiad

February 8, 2023

Taiwangeometry

Problem Statement

Let

O

be the center of circle

\Gamma

, and

A

,

B

be two points on

\Gamma

so that

O, A

and

B

are not collinear. Let

M

be the midpoint of

AB

. Let

P

and

Q

be points on

OA

and

OB

, respectively, so that

P \neq A

and

P, M, Q

are collinear. Let

X

be the intersection of the line passing through

P

and parallel to

AB

and the line passing through

Q

and parallel to

OM

. Let

Y

be the intersection of the line passing through

X

and parallel to

OA

and the line passing through

B

and orthogonal to

OX

. Prove that: if

X

is on

\Gamma

, then

Y

is also on

\Gamma

.
Proposed by usjl

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