Question

In a $ABC$, $E$ and $F$ are mid-points of sides $AB$ and $AC$ respectively. If $BF$ and $CF$ intersect each other at point $O$, Find If area(Triangle $\Delta$ OBC) =area(Quadrilateral $\square$AEOF).

Answer 1

Join E and F, Now E and F are the mid-points of AB and AC respectively.

Therefore, by mid point theorem

$EF\parallel BC$

Now, $ar△BEF=ar△CFE$ [area between same parallel lines]

Subtract $ar△EOF$ both circles,

$ar△BEF-ar△EOF=ar△CFE-ar△EOF$

$ar△BOE=ar△FOC\to \left(1\right)$

BF is median, therefore

$ar△ABF=ar△CFB\to \left(2\right)$

Subtract (1) from (2), we get

Now, $ar△ABF-ar△BOE=ar△CFB-ar△FOC$

$arquad.AEOF=ar△BOC$