Competition problem — Geometry — Maths Olympiad Prep

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Geometry Difficulty 4.5 AIME Find the answer

Suppose $A B C$ is a triangle such that $A B=13, B C=15$ , and $C A=14$ . Say $D$ is the midpoint of $\overline{B C}, E$ is the midpoint of $\overline{A D}, F$ is the midpoint of $\overline{B E}$ , and $G$ is the midpoint of $\overline{D F}$ . Compute the area of triangle $E F G$ .

Your answer

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

Solution

By Heron's formula, $[A B C]=\sqrt{21(21-15)(21-14)(21-13)}=84$ . Now, unwinding the midpoint conditions yields $[E F G]=\frac{[D E F]}{2}=\frac{[B D E]}{4}=\frac{[A B D]}{8}=\frac{[A B C]}{16}=\frac{84}{16}=\frac{21}{4}$ .

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