Question

prove that if a lines divides any two sides of a triangle in the same ratio then the line is parallel to the third side

Answer 1

Given ABC is a triangle E and F are midpoints of the sides AB ,AC respectively.

To prove : EF|| BC and  EF = ½ BC.

Construction : Draw a line CD parallel to AB ,it intersects EF at D.

Proof :

In a ΔAEF and ΔCDF

∠EAF = ∠FCD  ( Alternative interior angles)

AF = FC    ( F is the midpoint)

∠AFE = ∠CFD ( vertically opp. Angles)

ΔAEF ≅ ΔCDF  (ASA congruence property)

So that EF = DF and AE = CD  ( By CPCT )

BE = AE = CD

∴ BCDE  is parallelogram.

⇒ ED | | BC (Opposite sides of parallelogram are parallel)

⇒ EF | | BC

Answer 2

plz refer to this attachment