MathDB

Let

P

be an arbitrary point in the interior of triangle

\triangle ABC

. Lines

\overline{BP}

and

\overline{CP}

intersect

\overline{AC}

and

\overline{AB}

E

and

F

, respectively. Let

K

and

L

be the midpoints of the segments

BF

and

CE

, respectively. Let the lines through

L

and

K

parallel to

\overline{CF}

and

\overline{BE}

intersect

\overline{BC}

S

and

T

, respectively; moreover, denote by

M

and

N

the reflection of

S

and

T

over the points

L

and

K

, respectively. Prove that as

P

moves in the interior of triangle

\triangle ABC

, line

\overline{MN}

passes through a fixed point. Proposed by Ali Zamani

Problem Statement