Problem 5

A circle with center

O

passes through vertices

A

and

C

of triangle

ABC

and intersects segments

AB

and

BC

again at distinct points

K

and

N

, respectively. The circumcircles of

ABC

and

KBN

meet at exactly two distinct points

B

and

M

. Prove that

\angle OMB

is a right angle.

Step 1 of 4: Locate the radical center

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In plain words

Three circles carry three pairwise equal-power lines. The radical-axis theorem says those three lines must meet at one balancing point.

AC,KN,BM\text{ are concurrent at }X

Detailed analysis

Take the circumcircle of $ABC$ , the circumcircle of $KBN$ , and the given circle with center $O$ . Their pairwise radical axes are $BM$ , $KN$ , and $AC$ , respectively. By the radical-axis theorem these three lines are concurrent; call the point $X$ .