Question

If a line circle any two sides of a triangle in the same ratio then the line is parallel to the third side.

Answer 1

Given,

$\frac{AD}{DB}=\frac{AE}{EC}$...(A)

$D{E}^{\prime }\parallel BC$...(B) [B.P.T]

$\frac{AD}{DB}=\frac{A{E}^{\prime }}{E^{\prime }C}$...(C)

From (A) and (C), we get,

$\frac{AE}{EC}=\frac{A{E}^{\prime }}{E^{\prime }C}$

if $E^{\prime }=E$ then only the above condition satisfies.

$\therefore$ it implies that from (B)

$DE\parallel BC$

Hence proved.