Prove that the triangles formed by joining the mid – points of the three sides of a triangle consecutively are similar to their original triangle.

Let there be a ΔABC with the mid points D,E and F of sides AB, AC and BC.

In ΔADE and ΔABC,

D is the mid point of AB and E is the mid point of AC.

By Midpoint theorem,

⇒

⇒

⇒ DE = BF |(1)

Similarly in ΔBFD & ΔBCA,

DF = EC = AE |(2)

Similarly in ΔCFE & ΔCBA

EF = AD = DB |(3)

In ΔADE & ΔBDF,

AD = DB |D is mid point

BF = DE |From (1)

DF = EA |From (2)

Thus, ΔADE & ΔBDF are similar to each other by SSS Similarity Rule.

⇒ ΔADE~ΔBDF

Similarly,

ΔADE~ΔEFC

ΔDBF~ΔEFC

In ΔADE & ΔDEF,

AD = EF |From (3)

DE = DE

EA = DF |From (2)

⇒ ΔADE~ΔDEF

Thus, all the smaller triangles are similar to each other.