MathDB

Let

ABC

be an acute-angled triangle, let

M

be the midpoint of

BC

and let

H

be the foot of the

B

-altitude. Let

Q

be the circumcenter of

ABM

and let

X

be the intersection point between

BH

and the axis of

BC

.
Show that the circumcircles of the two triangles

ACM

AXH

and the line

CQ

pass through a same point if and only if

BQ

is perpendicular to

CQ

Problem Statement