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The line MN bisects the segment \overline{AH}

Source: 7th European Mathematical Cup , Junior Category, Q3

December 25, 2018

geometry

Problem Statement

Let

ABC

be an acute triangle with

|AB | < |AC |

and orthocenter

H

. The circle with center A and radius

|AC |

intersects the circumcircle of

\triangle ABC

at point

D

and the circle with center

A

and radius

|AB |

intersects the segment

\overline{AD}

at point

K.

The line through

K

parallel to

CD

intersects

BC

at the point

L.

If

M

is the midpoint of

\overline{BC}

and N is the foot of the perpendicular from

H

to

AL,

prove that the line

MN

bisects the segment

\overline{AH}

.

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