Competition problem — Geometry — Maths Olympiad Prep

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Geometry Difficulty 5.0 AIME Find the answer

Let $A X B Y$ be a cyclic quadrilateral, and let line $A B$ and line $X Y$ intersect at $C$ . Suppose $A X \cdot A Y=6, B X \cdot B Y=5$ , and $C X \cdot C Y=4$ . Compute $A B^{2}$ .

Your answer

A number or a short expression. Spacing and $ signs are ignored.

Solution

Observe that $\begin{aligned} & \triangle A C X \sim \triangle Y C B \Longrightarrow \frac{A C}{A X}=\frac{C Y}{B Y} \\ & \triangle A C Y \sim \triangle X C B \Longrightarrow \frac{A C}{A Y}=\frac{C X}{B X} \end{aligned}$ Mulitplying these two equations together, we get that $A C^{2}=\frac{(C X \cdot C Y)(A X \cdot A Y)}{B X \cdot B Y}=\frac{24}{5}$ Analogously, we obtain that $B C^{2}=\frac{(C X \cdot C Y)(B X \cdot B Y)}{A X \cdot A Y}=\frac{10}{3}$ Hence, we have $A B=A C+B C=\sqrt{\frac{24}{5}}+\sqrt{\frac{10}{3}}=\frac{11 \sqrt{30}}{15}$ implying the answer.

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