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Irish Mathematical Olympiad

Geometry Difficulty 4.3 AIME Prove it Ireland

The points $E$ and $F$ are on the sides $AC$ and $AB$ , respectively, of triangle $ABC$ such that $FE$ is parallel to $BC$ . The lines $BE$ and $CF$ intersect at $G$ . Prove that the line $AG$ passes through the midpoint of $BC$ .

Solution

Let $AG$ meet $BC$ at $D$ . We need to show that $D$ is the midpoint of $BC$ .

As $EF$ is parallel to $BC$ , we have $\frac{|CE|}{|EA|} = \frac{|BF|}{|FA|}$ . Ceva's Theorem tells us that
$\frac{|BD|}{|DC|} \cdot \frac{|CE|}{|EA|} \cdot \frac{|FA|}{|BF|} = 1.$
Together these imply $|BD| = |DC|$ , hence $D$ is the midpoint of $BC$ .

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