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 4 of 5: Identify the segment

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In plain words

At both ends of this phase the moving triangle coincides with a copy of $OAB$ or $OBC$ , so $X$ returns to $O$ ; continuity fills the segment between the extreme positions.

X=O\text{ at both endpoints}

Detailed analysis

At the initial and final configurations of this phase, $X=O$ . During the continuous motion it stays on the fixed line $BO$ , and the inside-polygon condition selects the segment from $O$ toward the corresponding vertex direction. Thus one phase traces one line segment with endpoint $O$ .