In triangle ABC, D, E and F are respectively the mid-points of sides AB, BC, CA. show that triangle ABC is divided into four congruent triangles by joining D, E, and F.
Given that D, E and F are the mid points of sides AB, BC, CA respectively.
To show: ΔABC is divided into four congruent triangles
Proof: D is the mid point of AB
F is the mid point of AC.
∴DF||BC (By mid point theorem)
also E is the mid point of BC
and F is the mid point of AC.
∴EF||AB (By mid point theorem)
By (1) & (2)
BEFD is a parallelogram
⇒ΔBDEΔDEF (Since diagonal of a parallelogram divides it into two congruent triangles) ......(3)
By (3), (4) & (5)