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 3 of 4: Write the two power identities

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In plain words

Power of a point turns geometry into products of two distances. Both products can be compared because the same radical-center configuration supplies them.

XM\cdot XB=XO^2-ON^2,\qquad BM\cdot BX=BO^2-ON^2

Detailed analysis

Since $X,M,N,C$ are cyclic, the intersecting chords/secants give $XM\cdot XB=XK\cdot XN$ . The latter is the power of $X$ with respect to the circle centered at $O$ , so it equals $XO^2-ON^2$ . Similarly, using the circumcircle through $B,K,N,M$ , the power of $B$ gives $BM\cdot BX=BN\cdot BC=BO^2-ON^2$ .