China Team Selection Test

Given a circle with radius 1 and two points $C$ and $D$ on it, and a constant $l$ with $0 < l \leq 2$. A moving chord $AB$ of the circle has length $l$, and $ABCD$ forms a non-degenerate convex quadrilateral. Let $AC$ and $BD$ intersect at $P$. We aim to find the loci of the circumcenters of triangles $ABP$ and $BCP$.

Let $T$ be the circumcenter of $\triangle BCP$. The angles $\angle TBC$ and $\angle TCB$ are fixed. Let $X$ be the second intersection of line $\overline{TB}$ with the circle, distinct from $B$. Since $C$ and $\angle XBC$ are fixed, $X$ is fixed. Consequently, the angle $\angle XTC = 2 \angle XBC$ is fixed. Therefore, $T$ lies on a circle passing through $X$ and $C$.

The loci of the circumcenters of triangles $ABP$ and $BCP$ are circles passing through fixed points determined by the configuration of the quadrilateral $ABCD$.

The answer is: $\boxed{\text{circles}}$ passing through fixed points}}.