D, E and F are respectively the mid-points of the sides AB, BC and CA of a triangle ABC. Prove that by joining these mid-points D, E and F, the triangles ABC is divided into four congruent triangles.

Question

Philoid · Accepted Answer

ABC is a triangle and D, E and F are mid-points of sides AB, BC and CA, respectively. Then,

AD = BD = AB

BE = EC = BC

And AF = CF = AC

Now, by mid-point theorem,

EF||AB and EF = AB = AD = BD

ED||AC and ED = AC = AF = CF

DF||BC and DF = BC = BE = CE

Now, in ΔADF and ΔEFD,

AD = EF

AF = DE

And DF = FD (common)

ADF EFD

Similarly, ΔDEF DEB

And ΔDEF CEF

Thus, ΔABC is divided into four congruent triangles.