In the figure the circle of radius $a$ is stationary, and for every $\theta$, the point $P$ is the midpoint of the segment $QR$. The curve traced out by $P$ for $0<\theta<\pi$ is called the longbow curve. Find the parametric equations for this curve.
Hint: The obvious choice as a parameter is $\theta$ as both $Q$ and $R$ depend on $\theta$. Try to express $R$ and $Q$ as functions of $\theta$, then notice that $P = \frac 1 2 (Q+R)$.
If you do not succeed, I suggest looking at the article Witch of Agnesi.