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HMMT February February 16, 2019

Geometry Difficulty 5.3 AIME, harder Prove it United States

Problem:

In rectangle $A B C D$ , points $E$ and $F$ lie on sides $A B$ and $C D$ respectively such that both $A F$ and $C E$ are perpendicular to diagonal $B D$ . Given that $B F$ and $D E$ separate $A B C D$ into three polygons with equal area, and that $E F=1$ , find the length of $B D$ .

Solution

Solution:

Observe that $A E C F$ is a parallelogram. The equal area condition gives that $B E = D F = \frac{1}{3} A B$ . Let $C E \cap B D = X$ , then $\frac{E X}{C X} = \frac{B E}{C D} = \frac{1}{3}$ , so that $B X^2 = E X \cdot C X = 3 E X^2 \Rightarrow B X = \sqrt{3} E X \Rightarrow \angle E B X = 30^{\circ}$ . Now, $C E = 2 B E = C F$ , so $C E F$ is an equilateral triangle and $C D = \frac{3}{2} C F = \frac{3}{2}$ . Hence, $B D = \frac{2}{\sqrt{3}} \cdot \frac{3}{2} = \sqrt{3}$ .

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