Math CentralQuandaries & Queries


Question from Dimitri, a student:

In the triangle ABC, M is the midpoint of AB and N is the midpoint of AC. Prove that CM and BN cannot bisect.
(I can prove it by contradiction if it is a scalene triangle, but i cant seem to prove it for isosceles)

I am not sure what 'bisect' means here. Here are two possible ways to look at the problem:

  1. If CM and BN meet in a common midpoint, then CNMB would be a parallelogram (that is, CN would have to be parallel to BM).

  2. Two medians CM and BN of a triangle meet in a point G (the centroid of the triangle) that lies 2/3 of the way from the vertex to the midpoint opposite side (so that CG = 2GM and BG = 2GN).


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