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Problem 2 -- Some Cyclic Points

Source: 46th Austrian Mathematical Olympiad National Competition Part 2 Problem 2

July 14, 2018

Austriageometrycyclic quadrilateral

Problem Statement

We are given a triangle

ABC

. Let

M

be the mid-point of its side

AB

.
Let

P

be an interior point of the triangle. We let

Q

denote the point symmetric to

P

with respect to

M

.
Furthermore, let

D

and

E

be the common points of

AP

and

BP

with sides

BC

and

AC

, respectively.
Prove that points

A

,

B

,

D

, and

E

lie on a common circle if and only if

\angle ACP = \angle QCB

holds.
(Karl Czakler)

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