Competition problem — Geometry — Maths Olympiad Prep

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Geometry Difficulty 4.6 AIME Find the answer

In triangle $A B C, \angle B A C=60^{\circ}$ . Let $\omega$ be a circle tangent to segment $A B$ at point $D$ and segment $A C$ at point $E$ . Suppose $\omega$ intersects segment $B C$ at points $F$ and $G$ such that $F$ lies in between $B$ and $G$ . Given that $A D=F G=4$ and $B F=\frac{1}{2}$ , find the length of $C G$ .

Your answer

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

Solution

Let $x=C G$ . First, by power of a point, $B D=\sqrt{B F(B F+F G)}=\frac{3}{2}$ , and $C E=\sqrt{x(x+4)}$ . By the law of cosines, we have $\left(x+\frac{9}{2}\right)^{2}=\left(\frac{11}{2}\right)^{2}+(4+\sqrt{x(x+4)})^{2}-\frac{11}{2}(4+\sqrt{x(x+4)})$ which rearranges to $2(5 x-4)=5 \sqrt{x(x+4)}$ . Squaring and noting $x>\frac{4}{5}$ gives $(5 x-16)(15 x-4)=0 \Longrightarrow x=\frac{16}{5}$ .

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