MathDB

Let

ABC

be an acute triangle such that

AH = HD

, where

H

is the orthocenter of

ABC

and

D \in BC

is the foot of the altitude from the vertex

A

. Let

\ell

denote the line through

H

which is tangent to the circumcircle of the triangle

BHC

. Let

S

and

T

be the intersection points of

\ell

with

AB

and

AC

, respectively. Denote the midpoints of

BH

and

CH

M

and

N

, respectively. Prove that the lines

SM

and

TN

are parallel.

Problem Statement