Prove or disprove by giving a counterexample.
If two medians of a triangle are of the same length,
then the triangle is isosceles.
Proof:
Draw triangle ABC with medians BD and CE, and label
their point of intersection X (which is the centroid).
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) Angle BXE = Angle CXD
5) Triangle BXE is congruent to Triangle CXD
6) BE = CD
7) 2(BE) = 2(CD)
8) AB = AC
Reasons:
1) Given
2) The medians of a triangle intersect
in a point that is 2/3 of the distance
from each vertex to the midpoint of
the opposite side.
3) Substitution property
4) Vertical angles are congruent
5) Side-angle-side
6) Corresponding parts of congruent triangles
are congruent
7) Multiplication property of equality
8) Midpoint theorem
Edwin
