Thread: Isosceles Triangle Proof

Prove that a triangle is isosceles if and only if two medians are congruent.

The basic idea behind this proof is that in two right triangles if the hypotenuses are congruent and two legs are congruent then the triangles are congruent.

I was able to use that idea to prove that a triangle is isosceles iff its altitudes are congruent but medians? What hypotenuse? I am having trouble with the part of the proof where I am assuming that the medians are congruent.

A different approach involves the fact that the centroid divides any median in the ratio 2:1. Suppose in triangle ABC, the medians AD and BE are equal and intersect at centroid G. Can you use the fact above to prove something about triangles AGE and BGD ?

posted by MATNTRNG
1 atriangleABC has a base BC and two rays AB andAC. midpoints of AB=M of AC =N
2 BN = CM given
3 MN is parallel to BC ( similar triangles )
4 arcs are drawn at B and C of equal lenghts ( median lenght)
5 a line parallel to BC is drawn above BC meeting arcs at M and N
6 MNCB is a trapezoid
7BN and CM are the trapezoid diagonals which are equal 4 above
8 MNCB is a regular trapezoid
9 a regular trapezoid is a truncated isosceles triangle
BM and CM extended meet at A the apex of isoscelesABC


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