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Altitude and midpoint circle

Source: Iran MO second round P4

April 19, 2024

geometry

Problem Statement

In the triangle

ABC

,

M

is the midpoint of

AB

and

B'

is the foot of

B

-altitude.

CB'M

intersects the line

BC

for the second time at

D

. Circumcircles of

CB'M

and

ABD

intersect each other again at

K

. The parallel to

AB

through

C

intersects the

CB'M

circle again at

L

. Prove that

KL

cuts

CM

in half.

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