Question

$D$ and $E$ are points on sides $AB$ and $AC$ respectively of $∆ABC$ such that $ar\left(DBC\right)=ar\left(EBC\right)$ . Prove that $DE\parallel BC$

Answer 1

Given that $ar\left(DBC\right)=ar\left(EBC\right)$

Both $∆DBC$and $∆EBC$ have common base $BC$.

We know that,

Area of triangle $=\frac{1}{2}\times base\times height$

Since base $BC$ is same for both $∆DBC$and $∆EBC$

their height must also be the same as areas are equal.

Therefore, the distance between lines $DE$ and $BC$ are exactly same which means $DE$ is parallel to $BC$.

Hence, $DE\parallel BC$