Doubling the Median

Doubling the Median

Theory

When proving the criterion for an isosceles triangle (If the bisector of a triangle coincides with its median, then the triangle is isosceles), the method of doubling the median is used. This auxiliary construction is also helpful in other problems involving medians.

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Problems

1. Prove that if two sides and the median to the third side of one triangle are equal respectively to two sides and the median to the third side of another triangle, then the triangles are congruent.

2. In triangle ABC, the median BM is drawn. It turns out that the sum of angles A and C is equal to angle ABM. Find the ratio of the median BM to side BC.

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