MathDB

Let

ABC

be an acute triangle with

AB<AC<BC, c

it's circumscribed circle and

D,E

be the midpoints of

AB,AC

respectively. With diameters the sides

AB,AC

, we draw semicircles, outer of the triangle, which are intersected by line

D

at points

M

and

N

respectively. Lines

MB

and

NC

intersect the circumscribed circle at points

T,S

respectively. Lines

MB

and

NC

intersect at point

H

. Prove that: a) point

H

lies on the circumcircle of triangle

AMN

b) lines

AH

and

TS

are perpedicular and their intersection, let it be

Z

, is the circimcenter of triangle

AMN

Problem Statement