Let AB be a fixed line segment. Let P be a moving point such that angle APB is equal to a constant acute angle. Then point P moves along which curve?

Question

Answer is: The boundary of union of two identical intersecting circles with centers outside the common region.

Answer 1

Let $P, P'$ be two such points. If $APP'B$ is cyclic, the $P'$ lies on the circumcircle of $\triangle APB$. Otherwise, the reflection of $P'$ across $\overline{AB}$, $P^*$, makes cyclic quadrilateral $APP^*B$, so $P'$ lies on the reflection of the circle across $AB$.