Apollonius's theorem
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.
This article is about the lengths of the sides of a triangle. For his work on circles, see Problem of Apollonius.Statement and relation to other theorem[edit]
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.
