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
