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circumcircles - cevians - midpoints, concurrency wanted

Source: Sharygin Geometry Olympiad 2017 First Round P19 grades 10-11

July 22, 2018

geometrycircumcircleconcurrencyCevianmidpoints

Problem Statement

Let cevians

AA', BB'

and

CC'

of triangle

ABC

concur at point

P.

The circumcircle of triangle

PA'B'

meets

AC

and

BC

at points

M

and

N

respectively, and the circumcircles of triangles

PC'B'

and

PA'C'

meet

AC

and

BC

for the second time respectively at points

K

and

L

. The line

c

passes through the midpoints of segments

MN

and

KL

. The lines

a

and

b

are defined similarly. Prove that

a

,

b

and

c

concur.

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