Question

If a line is drawn parallel to one side of a triangle to intersect the other two sides in distinct points then prove that the two sides are divided in the same ratio.

Answer 1

Given:

$DE||BC$

To prove that:

$\frac{EC}{AE}=\frac{BD}{AD}$

Proof:

$\angle AED=\angle ACB$ Corresponding angles

$\angle ADE=\angle ABC$ Corresponding angles

$\angle EAD$ is common to both the triangles

$\Rightarrow \Delta AED\sim \Delta ACB$ by $AAA$ similarity

$\Rightarrow \frac{AC}{AE}=\frac{AB}{AD}$

$\Rightarrow \frac{AE+EC}{AE}=\frac{AD+BD}{AD}$

$\Rightarrow \frac{EC}{AE}=\frac{BD}{AD}$

Hence proved