Prove that the medians to the legs of an isosceles triangle are congruent.

Step-by-step explanation:

Let ABC be an isosceles triangle with sides AC and BC of equal length.                

We need to prove that the medians AD and BE are of equal length.

Consider the triangles ADC and BEC.

They have two congruent sides that include congruent angles.

Indeed, AC = BC by the condition, because the triangle ABC is isosceles.

Since the lateral sides AC and BC are of equal length, their halves EC

and DC are of equal length too: EC = DC.

Finally, the angle ECD is the common angle.

Thus, the triangles ADC and BEC are congruent, in accordance to the

postulate P1 (SAS) (see the lesson Congruence tests for triangles of the

topic Triangles in the section Geometry in this site).

Hence, the medians AD and BE are of equal length as the corresponding sides

of these triangles.

The proof is completed.