Question

Prove that the medians of an equilateral triangle are equal.

Answer 1

Let the $\Delta ABC$ be equilateral triangle with medians $AE,FB,CD$.

Consider the triangles $\Delta ABF$ and $\Delta BAE$.

Since $AC=BC$
$\Rightarrow \frac{AC}{2}=\frac{BC}{2}$ [Dividing $2$ on both sides]
Thus, $AF=BE$
$\angle FAB=\angle EBA$ [ Since, in equilateral triangle, all three are equal]
$AB=AB$ (common side)
Thus, by SAS criterion, we get $\Delta ABF\cong \Delta BAE$
Therefore, $AE=BF$ ........(i)
Similar way, we can prove $CD=AE$..........(ii), by taking $AB=BC$
From equations(i) and (ii), we get $AE=BF=CD$
Hence, the the medians of equilateral triangle are equal.