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.

In any triangle

{\displaystyle ABC,}

{\displaystyle AD}

is a median, then

It is a special case of Stewart's theorem.

For an isosceles triangle with

is perpendicular to

and the theorem reduces to the Pythagorean theorem for triangle

{\displaystyle ADB}

(or triangle

From the fact that the diagonals of a parallelogram bisect each other, the theorem is equivalent to the parallelogram law.

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

be the length of the segments of

formed by the median, so

Let the angles formed between

includes

= − cos ⁡ θ .

The law of cosines for

states that

− 2 d m cos ⁡ θ

+ 2 d m cos ⁡ θ .

Add the first and third equations to obtain