HMMT November

Geometry Difficulty 4.4 AIME Find the answer

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 $A D$ .

Your answer

A number or a short expression. Fractions can be typed as 3/2, and spacing doesn't matter.

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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