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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:
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:
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:
Denote the length of the median AM by ma, and the lengths of the sides of the triangle by:
Apply the law of cosines in the triangle ABC:
as well as in the triangle DAC:
we can add the two equalities to obtain:
and we can solve for the length of the median onto BC:
The lengths of the other two medians can be obtained by permuting the sides circularly, since the proof is similar:
This computation is often useful in problems where other techniques lead to longer solution paths.