Problem 1765 (difficulty: 5/10)

Show that there is exactly one conformal map which

(a) takes a given circle \(\displaystyle C\) to another circle \(\displaystyle C'\) in such a way that it takes 3 prescribed points on \(\displaystyle C\) to 3 prescribed points on \(\displaystyle C'\).

(b) takes a given circle \(\displaystyle C\) to another circle \(\displaystyle C'\) in such a way that it takes a prescribed point on \(\displaystyle C\) to a prescribed point on \(\displaystyle C'\) and a prescribed point inside \(\displaystyle C\) to a prescribed point inside \(\displaystyle C'\).

Give me another random problem!