green/blue areas = red area

Pythagoras as a special case:
green area = red area

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".

Specifically, 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.

The theorem is named for the ancient Greek mathematician Apollonius of Perga.

Proof[]

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[]

• Formulas involving the medians' lengths – Line segment joining a triangle's vertex to the midpoint of the opposite side

References[]

External links[]

• Apollonius Theorem at PlanetMath.
• David B. Surowski: Advanced High-School Mathematics. p. 27

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