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(Redirected from Apollonius' theorem)
Relates the length of a median of a triangle to the lengths of its sides
This article is about the lengths of the sides of a triangle. For his work on circles, see Problem of Apollonius.
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.
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.
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.