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 1 of 5: Analyze one side-to-side phase

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In plain words

It is enough to understand what happens while the moving triangle passes from the copy of $OAB$ to the copy of $OBC$ ; all other phases are rotations of this one.

\angle YXZ=\angle AOB=\frac{2\pi}{n},\quad\angle YBZ=\angle ABC=\pi-\frac{2\pi}{n}

Detailed analysis

Let $C\ne A$ be the vertex adjacent to $B$ . During the phase from $OAB$ to $OBC$ , the angle at $X$ is $\angle YXZ=\angle AOB=\frac{2\pi}{n}$ , while the polygon angle at $B$ is $\angle YBZ=\angle ABC=\pi-\frac{2\pi}{n}$ .