Apollonius' median theorem

If we draw the median to one side of a traingle then we get the following result.

This result can be proved in (at least) two ways.

The first involves drawing the altitude of A on BC (ie dropping a perpendicular). This then gives two right angles triangles and the result drops out by using Pythagoras with AB abd AC as the hypotenuses.

The alternative is to use the cosine rule to find both AB abd AC, remembering that cos(180-θ)=-cosθ.

Hints to both methods are given in the diagram below.

To download this Geogebra file click here.