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fixed point for the circumcircle, equal angles (Kyiv City Olympiad 2007 11.5)

Source:

June 29, 2020

geometrycircumcircleequal anglesFixed pointfixed

Problem Statement

The points

A

and

P

are marked on the plane. Consider all such points

B, C

of this plane that

\angle ABP = \angle MAB

and

\angle ACP = \angle MAC

, where

M

is the midpoint of the segment

BC

. Prove that all the circumscribed circles around the triangle

ABC

for different points

B

and

C

pass through some fixed point other than the point

A

.
(Alexei Klurman)