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Berkeley Math Circle Monthly Contest 1

Geometry Difficulty 4.6 AIME Prove it United States

Problem:

In the interior of a triangle $ABC$ with area $1$ , points $D$ , $E$ , and $F$ are chosen such that $D$ is the midpoint of $AE$ , $E$ is the midpoint of $BF$ , and $F$ is the midpoint of $CD$ . Find the area of triangle $DEF$ .

Solution

Solution:

Let $x$ be the area of $\triangle DEF$ . Comparing triangles $ABE$ and $DEF$ , we find that base $AE$ is twice base $DE$ but, since $E$ bisects $BF$ , the heights to these bases are equal. Thus $\triangle ABE$ has area $2x$ . Symmetrically, triangles $BCF$ and $CAD$ have area $2x$ . Since these four triangles fill up $\triangle ABC$ , we have $1 = x + 2x + 2x + 2x = 7x$ , so $x = 1/7$ .

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