In geometry, Apollonius's theorem is a theorem relating the length of a median of a triangle to the lengths of its sides. It states that the sum of the squares of any two sides of any triangle equals twice the square on half the third side, together with twice the square on the median bisecting the third side. The theorem is named for the ancient Greek mathematician Apollonius of Perga.

green/blue areas = red area
Pythagoras as a special case:
green area = red area

Statement and relation to other theorem

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In any triangle   if   is a median, then   It is a special case of Stewart's theorem. For an isosceles triangle with   the median   is perpendicular to   and the theorem reduces to the Pythagorean theorem for triangle   (or triangle  ). From the fact that the diagonals of a parallelogram bisect each other, the theorem is equivalent to the parallelogram law.

Proof

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Proof of Apollonius's theorem

The theorem can be proved as a special case of Stewart's theorem, or can be proved using vectors (see parallelogram law). The following is an independent proof using the law of cosines.[1]

Let the triangle have sides   with a median   drawn to side   Let   be the length of the segments of   formed by the median, so   is half of   Let the angles formed between   and   be   and   where   includes   and   includes   Then   is the supplement of   and   The law of cosines for   and   states that  

Add the first and third equations to obtain   as required.

See also

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References

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  1. ^ Godfrey, Charles; Siddons, Arthur Warry (1908). Modern Geometry. University Press. p. 20.
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