Question

If $D$ and $E$ are points on side $AB$ and $AC$ respectively of triangle $ABC$ such that $ar\left(\Delta DBC\right)=ar\left(\Delta EBC\right)$, then prove that $DE\parallel BC$

Answer 1

If Area $\left(△DBC\right)=$ Area $\left(△EBC\right)$

then

$\frac{1}{2}|BC||DF|=\frac{1}{2}|BC||EG|$

$|DF|=|EG|$

Also $EG\perp BC$ & $DF\perp BC$

then

$D$ & $E$ are equidistant from $BC$

and $DE\parallel BC$ because line joining equidistant point from a line is $\parallel$ to that line