This statement is usually formulated as the 

Theorem. In an isosceles triangle, the angle bisector to the angle between congruent sides is the median. 


See the lesson An altitude a median and an angle bisector in the isosceles triangle in this site.


Below is the proof from this lesson.


  

  

Figure 3 shows an isosceles triangle ABC with sides AB and AC of equal length.
The segment AD is the bisector of the angle BAC opposite to the base BC.
We need to prove that CD is the median of the triangle ABC.


Since AD is a bisector of the angle BAC, the angles BAD and CAD are congruent.
Thus, the triangles ADB and ADC have the pair of congruent sides AB = AC, the common                      
side AD and the congruent included angles BAD and CAD.
Hence, these triangles are congruent, in accordance to the postulate P1 (SAS) of the triangle 
congruency (see the lesson  Congruence tests for triangles under the current topic in this site).
It implies that the segments BD and CD are congruent. 
This means that the bisector of the angle ACB is the median. 
Thus, the statement is proved.
 
  



        Figure