In cyclic quadrilateral $ABCD$, let $AD$ and $CB$ meet at $P$. Let $E$ and $F$ be the midpoints of $DB$ and $CA$ respectively. Find $\angle PEF$ in terms of $\alpha, \beta, \gamma, \delta$, which are the angles of the cyclic quadrilateral.
Angle $\angle PEF$, in general, does not depend only on the angles of the cyclic quadrilateral. See below an example, with two cyclic quadrilaterals having the same angles (they have parallel sides) but different values for $\angle PEF$. You also need to know the radius of the circle.
The radius of the circle is not enough: it is clear that one can scale the second figure so as to make two equal circles while keeping the same angles.