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 2 of 4: Use an angle chase to find a second cyclic quadrilateral

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In plain words

The first three circles create a new four-point circle: angle equality is the signal that the fourth point has joined the same circle.

X,M,N,C\text{ are concyclic}

Detailed analysis

Because $A,C,K,N$ lie on the given circle and $B,K,M,N$ lie on the second circumcircle, compare angles at $M$ and $C$ . Depending on the order of the collinear points, the directed-angle chase gives $\angle XMN=180^\circ-\angle XCN$ (with the equivalent supplementary-angle form in the other arrangement). Hence $X,M,N,C$ are concyclic.