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circumcircle k of ABC passes through P iff k passes through midpoint of EF

Source: 2010 Balkan Shortlist G6 BMO

April 4, 2020

geometrycircumcircle

Problem Statement

In a triangle

ABC

the excircle at the side

BC

touches

BC

in point

D

and the lines

AB

and

AC

in points

E

and

F

respectively. Let

P

be the projection of

D

on

EF

. Prove that the circumcircle

k

of the triangle

ABC

passes through

P

if and only if

k

passes through the midpoint

M

of the segment

EF

.

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