Let $P$ be a point inside circle $C_1$.Consider the set of chords of $C_1$ that contain $P$. Prove that their midpoints all lie on a circle.
My observation: Let $O$ be the centre of $C_1$.$O$ must in the required circle. To show some points concyclic usually we need angles.I tried to find angled at the midpoints but failed to find any angle. Please help me. Thank you!
Guess It is just my guess but constructing a good diagram I found that the circle may be the circle with diameter $OP$.