24. Four points in the order $A, B, C, D$ lie on a circle with the extension of $A B$ meeting the extension of $D C$ at $E$ and the extension of $A D$ meeting the extension of $B C$ at $F$. Let $E P$ and $F Q$ be tangents to this circle with points of tangency $P$ and $Q$ respectively. Suppose $E P=60$ and $F Q=63$. Determine the length of $E F$.
24. Ans: 87
Let the circumcircle of $\triangle C D F$ meet the line $E F$ at $G$. Then $G, C, D, F$ are concyclic. Now $\angle E B C=\angle A D C=\angle C G F$ so that $E, B, C, G$ are also concyclic. Thus, $E P^{2}=E B \cdot E A=E C \cdot E D=E G \cdot E F$ and $F Q^{2}=F D \cdot F A=F C \cdot F B=F G \cdot F E$. Therefore, $E P^{2}+F Q^{2}=E G \cdot E F+F G \cdot F E=(E G+F G) \cdot E F=E F^{2}$. Consequently, $E F=\sqrt{60^{2}+63^{2}}=87$.