Math CentralQuandaries & Queries


Question from Nazrul, a teacher:

If the three medians of a triangle are equal, how can I prove (without using the property that the three medians of a triangle cut one another at a point of trisection) that the triangle is equilateral.

Try drawing an accurate picture of a triangle with its medians. Then through the ends of one median draw lines parallel to the other two medians. Specifically, if you label your triangle ABC and the midpoints of the opposite sides D, E, and F, determine the point E' so that DE' is parallel to BE and AE' is parallel to FC. The sides of the triangle ADE' are then equal to the three medians, so that if you start with equal medians, then triangle ADE' is equilateral. Now look in your picture how the lines E'E, AE, and DE are related to the sides of the starting triangle ABC. All that is left for you to do is to prove that what you see is actually true.


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