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This essential trains for: AMC-12, AIME.

The problem is to calculate the length of a median in a triangle, if the lengths of the sides are given. In the figure, let M be the midpoint of the segment BC:

figure

To do this, extend the median from its point of intersection with the opposite vertex further outside the triangle, by a segment equal to its own length, like in the figure:

figure

equation

Notice how the segments BC and AD bisect each other. This means that ABCD is a parallelogram with these segments as diagonals. In this case:

equation

Denote the length of the median AM by ma, and the lengths of the sides of the triangle by:

equation

equation

equation

Apply the law of cosines in the triangle ABC:

equation

as well as in the triangle DAC:

equation

figure

Since:

equation

we can add the two equalities to obtain:

equation

and we can solve for the length of the median onto BC:

equation

The lengths of the other two medians can be obtained by permuting the sides circularly, since the proof is similar:

equation

equation

This computation is often useful in problems where other techniques lead to longer solution paths.