D is the mid point of side $BC$ of $Δ ABC$ and $E$ is the mid point of $BD$ . If $O$ is the mid point of $AE$ , prove that $ar Δ BOE = \frac{1}{8} ar Δ ABC$

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Solution

Since $A D$ and $A E$ are median of $△ A B C$ and $△ A B D$ respectively.
$∴ a r (△ A B D) = \frac{1}{2} a r (△ A B C) . . . . . (i)$
and $a r (△ A B E) = \frac{1}{2} a r (△ A B D) . . . . (i i)$
$O B$ is a median of $△ A B E$
$∴ a r (△ B O A) = \frac{1}{2} a r (△ A B E) . . . (i i i)$
From $(i), (i i)$ , we have
$a r (△ B O E) = \frac{1}{8} a r (△ A B C)$

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