Let $A,B$ be adjacent vertices of a regular $n$-gon ($n\ge5$) with center $O$. A triangle $XYZ$, which is congruent to and initially coincides with $OAB$, moves in the plane in such a way that $Y$ and $Z$ each trace out the whole boundary of the polygon, with $X$ remaining inside the polygon. Find the locus of $X$.
Let $ C \not = A$ the vertex which is adjacent to $ B$. While $ XYZ$ moves from $ OAB$ to $ OBC$, it is easy to see $ XYBZ$ is cyclic. Thus $ X$ lies on the bisector of $ \angle YBZ = \angle ABC$. Moreover, $ X$ is the intersection of a circle passing through $ B$ (the circumcircle of $ XYBZ$) and with a fixed radius (the radius is a function of $ \triangle XYZ$). Therefore $ X$ varies in a line segment ended in $ O$. When $ Y$ and $ Z$ pass through the other sides, we get as locus $ n$ distinct line segments, each passing throught $ O$ and contained in $ OV$ (but not in $ \vec{OV}$) for some vertex $ V$ of the polygon. Each two of these lines are obtained one from another by a rotation with center $ O$.