MathDB

Let ABC be a triangle with

|AB|< |AC|.

Let

k

be the circumcircle of

\triangle ABC

and let

O

be the center of

k

. Point

M

is the midpoint of the arc

BC

k

not containing

A

. Let

D

be the second intersection of the perpendicular line from

M

AB

with

k

and

E

be the second intersection of the perpendicular line from

M

AC

with

k

. Points

X

and

Y

are the intersections of

CD

and

BE

with

OM

respectively. Denote by

k_b

and

k_c

circumcircles of triangles

BDX

and

CEY

respectively. Let

G

and

H

be the second intersections of

k_b

and

k_c

with

AB

and

AC

respectively. Denote by ka the circumcircle of triangle

AGH.

Prove that

O

is the circumcenter of

\triangle O_aO_bO_c,

where

O_a, O_b, O_c

are the centers of

k_a, k_b, k_c

respectively.

Problem Statement