Question
$D$ and $E$ are points on sides $AB$ and $AC$ respectively of $\triangle\text{ABC}$ such that$\text{ar}(\triangle\text{BCD})=\text{ar}(\triangle\text{BCE}).$ Prove that $DE \| BC.$
✓
Answer
Since $\triangle\text{BCD}$ and $\triangle\text{BCE}$ are equal in area and have a same base $BC$.
Therefore, Altitude from $D$ of $\triangle\text{BCD}$ = Altitude from $E$ of $\triangle\text{BCE}.$
$\Rightarrow\ \triangle\text{BCD}$ and $\triangle\text{BCE}.$ are between the same parallel lines.
$\Rightarrow DE \| BC.$
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