Question

Prove that If a line divides any two sides of a triangle in the same ratio,then the line is parallel to same side.

Answer 1

Let ABC be a triangle with BC as its base.

Denote D as the mid point of AB,

Denote E as the mid point of AC.

Join CD and BE.

In $△ADE,△BDE$

$\frac{Ar△ADE}{Ar△BDE}=\frac{AD}{BD}$

In $△ADE,△CDE$

$\frac{Ar△ADE}{Ar△CDE}=\frac{AE}{CE}$

Since the base DE is same, we have,

$Ar△BDE=Ar△CDE$

$\therefore \frac{Ar△ADE}{Ar△BDE}=\frac{Ar△ADE}{Ar△CDE}$

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

Hence proved.