91Ó°ÊÓ

The diagram is:

From the diagram it can be noticed that the triangle $ABC$is an isosceles triangle having sides $AB$and $AC$as the legs of the isosceles triangle. $BD$androle="math" localid="1650432749160" $CE$ are the medians to the legs of isosceles triangle.

As, the triangle$ABC$is an isosceles triangle, therefore $AB=AC$.

The angle opposite to the equal sides are also equal, therefore $∠ABC=∠ACB$.

As,$D$is the midpoint of $AC$, therefore by using the definition of midpoint it can be said that $CD=\frac{1}{2}AC$.

As,$E$is the midpoint of $AB$, therefore by using the definition of midpoint it can be said that $BE=\frac{1}{2}AB$.

As, $AB=AC$, therefore it can be noticed that:

$\begin{array}{c}AB=AC\\ \frac{1}{2}AB=\frac{1}{2}AC\\ BE=CD\end{array}$

In the triangles$â–³BEC$ and $â–³CBD$, it can be seen that $BEâ‰\dots CD$,$∠ABCâ‰\dots ∠ACB$ and $BCâ‰\dots BC$.

Therefore, the triangles$â–³BEC$ and$â–³CDB$are the congruent triangles by using SAS postulate.

Therefore, by using the corresponding parts of congruent triangles, it can be said that $CEâ‰\dots BD$.

Hence proved that $BDâ‰\dots CE$.

Hence proved that the medians drawn to the legs of an isosceles triangle are congruent.

The proof in two-column form is: