Question

If the length of three medians of a triangle are equal, Let us prove that the triangle is an isosceles triangle.

Answer 1

In $△BGD$ and $△CGE$,
$\angle BGD=\angle CGE$ (vertically opposite angles) ...(1)
$BE=DC$ (medians are equal)
Since, centroid divides the median in ratio 2:1
So, $BG=\frac{2}{3}BE$ and $CG=\frac{2}{3}CD$
$\Rightarrow BG=CG($ as $BE=DC)$
And $GE=\frac{1}{3}BE$ and $DG=\frac{1}{3}CD$
$\Rightarrow GE=DG(BE=DC)$
Hence, by SAS congruency, $△BGD$ and $△CGE$ are congruent.
Hence, by SAS congruency, $△BGD$ and $△CGE$ are congruent.
By $CPCT,BD=EC$
$2\times BD=2\times EC$
$\Rightarrow AB=AC$
Hence, the triangle is isosceles.