MathDB

Let

A

and

B

different points of a circle

k

centered at

O

in such a way such that

AB

is not a diagonal of

k

. Furthermore, let

X

be an arbitrary inner point of the segment

AB

. Let

k_1

be the circle that passes through the points

A

and

X

, and

A

is the only common point of

k

and

k_1

. Similarly, let

k_2

be the circle that passes through the points

B

and

X

, and

B

is the only common point of

k

and

k_2

. Let

M

be the second intersection point of

k_1

and

k_2

. Let

Q

denote the center of circumscribed circle of the triangle

AOB

. Let

O_1

and

O_2

be the centers of

k_1

and

k_2

. Show that the points

M,O,O_1,O_2,Q

are on a circle.

Problem Statement