Question

$D,E$ and $F$ are the mid-points of the sides $AB,BC$ and $CA$ of an isosceles $\Delta ABC$ in which $AB=BC$. Prove that $\Delta DEF$ is also isosceles.

Answer 1

Given : D,E,F are the midpoints of $AB,BC,CA$

To prove : $DE=EF$

Proof :$AB=AC$ (isoscles triangle)

$\angle B=\angle C$ (angles opp to equal sides)

$\frac{1}{2}AB=\frac{1}{2}AC$

so, $DB=CF$

In $\Delta ADF$ and $\Delta CEF$

$DB=CF$

$DE=CF(E$ is midpoint of $BC)$

$\angle B=\angle C$ (proved above)

$\therefore \Delta ADF=\Delta CEF$ (SAS)

$DE=FE$ (CPCT)

Hence proved