Q. 73.6( 10 Votes )

# Prove that

Answer :

Let triangle be ∆*ABC*. *D, E* and *F* are the midpoints of sides *AB, BC* and *CA*, respectively.

**By midpoint** theorem, for D and E as midpoints of sides AB and BC,

*DE* *∣∣* *AC*

*Similarly, DF* *∣∣* *BC* and *EF* ***∣∣* *AB*.

∴ *ADEF, BDFE* and *DFCE* are all parallelograms.

*But, DE* is the diagonal of the *BDFE*.

∴ ∆*BDE* *≅* ***∆**FED* *…**(1)*

Similarly, DF is the diagonal of the parallelogram ADEF.

∴ ∆DAF ≅ ∆FED *…(2)*

And, EF is the diagonal of the parallelogram DFCE.

∴ ∆EFC ≅ ∆FED …(3)

Hence, all the four triangles are congruent.

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