KURSCHAK 2003 Problem 1

Draw a circle $k$ with diameter $\overline{EF}$ , and let its tangent in $E$ be $e$ . Consider all possible pairs $A,B\in e$ for which $E\in \overline{AB}$ and $AE\cdot EB$ is a fixed constant. Define $(A_1,B_1)=(AF\cap k,BF\cap k)$ . Prove that the segments $\overline{A_1B_1}$ all concur in one point.