# Apollonius's theorem

In geometria, 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".

# Apollonius's theorem

green/blue areas = red area

Pythagoras as a special case:
area verde = area rossa

In geometria, Apollonius's theorem è un teorema relating the length of a median of a triangolo 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".

In particolare, in any triangle

$"{displaystyle$

Se

$"{displaystyle$

is a median, poi

$"{displaystyle$=2left(|ANNO DOMINI|^{2}+|BD|^{2}Giusto).}

It is a special case di Stewart's theorem. For an isosceles triangle insieme a

$"{displaystyle$

the median

$"{displaystyle$

is perpendicular to

$"{displaystyle$

and the theorem reduces to the Pythagorean theorem for triangle

$"{displaystyle$displaystyle ADB}

(or triangle

$"{displaystyle$displaystyle ADC}

). 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.

Índice

## Prova[]

Proof of Apollonius's theorem

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

Let the triangle have sides

$"{displaystyle$

with a median

$"{displaystyle$

drawn to side

$"{displaystyle$displaystyle a.}

Permettere

$"{displaystyle$

be the length of the segments of

$"{displaystyle$

formed by the median, Così

$"{displaystyle$

is half of

$"{displaystyle$displaystyle a.}

Let the angles formed between

$"{displaystyle$

e

$"{displaystyle$

be

$"{displaystyle$

e

$"{displaystyle$displaystyle theta ^{primo },}

dove

$"{displaystyle$

includes

$"{displaystyle$

e

$"{displaystyle$displaystyle theta ^{primo }}

includes

$"{displaystyle$displaystyle c.}

Quindi

$"{displaystyle$displaystyle theta ^{primo }}

is the supplement of

$"{displaystyle$

e

$"{displaystyle$displaystyle cos theta ^{primo }=-cos theta .}

Il law of cosines per

$"{displaystyle$

e

$"{displaystyle$displaystyle theta ^{primo }}

afferma che

$"{displaystyle$=m^{2}+d^{2}-2dmcos theta \c^{2}&=m^{2}+d^{2}-2dmcos theta '\&=m^{2}+d^{2}+2dmcos theta .,end{allineato}}}

Add the first and third equations to obtain

$"{displaystyle$

come richiesto.

## Riferimenti[]

1. ^

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