Prove that if 2 medians of a triangle are congruent 
then the triangle is isosceles

Construct Triangle ABC with medians BD and CE, 
and the centroid (intersection of the medians) as X.

Given: Medians BD and CE, and BD = CE
Prove: AB = AC

Statements:
1) Medians BD and CE, and BD = CE
2) BX = (2/3)(BD), DX = (1/3)(BD), CX = (2/3)(CE), EX = (1/3)(CE)
3) BX = CX, DX = EX
4) mÐBXE = mÐCXD 
5) D BXE @ D CXD
6) BE = CD
7) 2(BE) = 2(CD)
8) BD = CE
Reasons:
1) Given
2) The medians of a triangle intersect in a point 
   that is two-thirds of the distance from each 
   vertex to the midpoint of the opposite side.
3) Substitution property
4) Vertical angles are congruent
5) SAS postulate
6) Corresponding parts of congruent triangles are 
   congruent
7) Multiplication property of equality 
8) Midpoint theorem

Edwin