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Proof of being Circumcenter of APQ from JMO 2010

Source: 2010 Japan Mathematical Olympiad Finals, Problem 1

February 11, 2010

geometrycircumcircleincenterperpendicular bisectorgeometry proposed

Problem Statement

Given an acute-angled triangle

ABC

such that

AB\neq AC

. Draw the perpendicular

AH

from

A

BC

. Suppose that if we take points

P,\ Q

in such a way that three points

A,\ B,\ P

and three points

A,\ C,\ Q

are collinear in this order respectively, then we have four points

B,\ C,\ P,\ Q

are concyclic and HP \equal{} HQ. Prove that

H

is the circumcenter of

\triangle{APQ}