HMMT November

Geometry Difficulty 4.5 AIME Prove it United States

Problem:

Points $A$ , $B$ , $C$ , $D$ lie on a circle in that order such that $\frac{A B}{B C}=\frac{D A}{C D}$ . If $A C=3$ and $B D=B C=4$ , find $A D$ .

Solution

Solution:

By Ptolemy's theorem, we have $A B \cdot C D + B C \cdot D A = A C \cdot B D = 3 \cdot 4 = 12$ .

Since the condition implies $A B \cdot C D = B C \cdot D A$ , we have $D A = \frac{6}{B C} = \frac{3}{2}$ .

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Source: MathNet, licensed CC-BY-4.0. Statement reproduced verbatim; metadata (topic, difficulty) added by this project.