Problem 4

Let

A,B

be adjacent vertices of a regular

n

-gon (

n\ge5

) with center

O

. A triangle

XYZ

, congruent to and initially coinciding with

OAB

, moves so that

Y

and

Z

each trace the whole boundary of the polygon, while

X

remains inside the polygon. Find the locus of

X

Step 2 of 5: Obtain a cyclic quadrilateral

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In plain words

Opposite angles summing to $\pi$ are exactly the cyclicity criterion.

\angle YXZ+\angle YBZ=\pi

Detailed analysis

The two displayed angles sum to $\pi$ , so $X,Y,B,Z$ lie on one circle throughout this phase. Since $XY=XZ$ by congruence, equal chords $XY$ and $XZ$ subtend equal angles at $B$ .