HMMT November

Geometry Difficulty 4.8 AIME Find the answer

In equilateral triangle $A B C$ , a circle $\omega$ is drawn such that it is tangent to all three sides of the triangle. A line is drawn from $A$ to point $D$ on segment $B C$ such that $A D$ intersects $\omega$ at points $E$ and $F$ . If $E F=4$ and $A B=8$ , determine $|A E-F D|$ .

Your answer

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

Solution

Without loss of generality, $A, E, F, D$ lie in that order. Let $x=A E, y=D F$ . By power of a point, $x(x+4)=4^{2} \Longrightarrow x=2 \sqrt{5}-2$ , and $y(y+4)=(x+4+y)^{2}-(4 \sqrt{3})^{2} \Longrightarrow y=\frac{48-(x+4)^{2}}{2(x+2)}=\frac{12-(1+\sqrt{5})^{2}}{\sqrt{5}}$ . It readily follows that $x-y=\frac{4}{\sqrt{5}}=\frac{4 \sqrt{5}}{5}$ .

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