MathDB

Let

ABC

be a triangle with

\angle ABC = 90^{\circ}

, and let

H

be the foot of the altitude from

B

. The points

M

and

N

are the midpoints of the segments

AH

and

CH

, respectively. Let

P

and

Q

be the second points of intersection of the circumcircle of the triangle

ABC

with the lines

BM

and

BN

, respectively. The segments

AQ

and

CP

intersect at the point

R

. Prove that the line

BR

passes through the midpoint of the segment

MN

Problem Statement