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concyclic wanted, circumcenters, reflection wrt line related

Source: 2019 Irish Mathematical Olympiad paper 1 p5

October 5, 2020

geometryreflectionConcyclicCircumcenter

Problem Statement

Let

M

be a point on the side

BC

of triangle

ABC

and let

P

and

Q

denote the circumcentres of triangles

ABM

and

ACM

respectively. Let

L

denote the point of intersection of the extended lines

BP

and

CQ

and let

K

denote the reflection of

L

through the line

PQ

. Prove that

M, P, Q

and

K

all lie on the same circle.