Competition problem — Geometry — Maths Olympiad Prep

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Geometry Difficulty 5.0 AIME Find the answer

$A B C$ is a triangle with points $E, F$ on sides $A C, A B$ , respectively. Suppose that $B E, C F$ intersect at $X$ . It is given that $A F / F B=(A E / E C)^{2}$ and that $X$ is the midpoint of $B E$ . Find the ratio $C X / X F$ .

Your answer

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

Solution

Let $x=A E / E C$ . By Menelaus's theorem applied to triangle $A B E$ and line $C X F$ , $1=\frac{A F}{F B} \cdot \frac{B X}{X E} \cdot \frac{E C}{C A}=\frac{x^{2}}{x+1}$ Thus, $x^{2}=x+1$ , and $x$ must be positive, so $x=(1+\sqrt{5}) / 2$ . Now apply Menelaus to triangle $A C F$ and line $B X E$ , obtaining $1=\frac{A E}{E C} \cdot \frac{C X}{X F} \cdot \frac{F B}{B A}=\frac{C X}{X F} \cdot \frac{x}{x^{2}+1}$ so $C X / X F=\left(x^{2}+1\right) / x=\left(2 x^{2}-x\right) / x=2 x-1=\sqrt{5}$ .

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