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The intersection lies on a circle

Source: 2023 China South-east Mathematical Olympiad Grade 10 P5

July 31, 2023

geometry

Problem Statement

Let

AB

be a chord of the semicircle

O

(not the diameter).

M

is the midpoint of

AB

, and

D

is a point lies on line

OM

(

D

is outside semicircle

O

). Line

l

passes through

D

and is parallel to

AB

.

P, Q

are two points lie on

l

and

PO

meets semicircle

O

at

C

. If

\angle PCD=\angle DMC

, and

M

is the orthocentre of

\triangle OPQ

. Prove that the intersection of

AQ

and

PB

lies on semicircle

O

.

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