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Circles tangent at orthocenter

Source: APMO 2018 P1

June 24, 2018

geometryAPMO

Problem Statement

Let

BMNC

be the orthocenter of the triangle

ABC

. Let

M

and

N

be the midpoints of the sides

AB

and

AC

, respectively. Assume that

H

lies inside the quadrilateral

BMNC

and that the circumcircles of triangles

BMH

and

CNH

are tangent to each other. The line through

MK

parallel to

BC

intersects the circumcircles of the triangles

BMH

and

MHN

in the points

K

and

L

, respectively. Let

F

be the intersection point of

MK

and

NL

and let

J

be the incenter of triangle

MHN

. Prove that

F J = F A

.

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