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 5 of 5: Apply

n

-fold symmetry

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In plain words

Each edge transition is obtained from the first by rotating through $2\pi/n$ about $O$ .

\mathcal L=\bigcup_{j=0}^{n-1}R_{2\pi j/n}(\overline{OV})

Detailed analysis

As $Y,Z$ pass all $n$ sides, rotational symmetry gives $n$ distinct congruent line segments, each passing through $O$ and lying along the line joining $O$ to a vertex (with only the inside-polygon half selected). Hence the locus is an $n$ -armed star of segments through $O$ .