Dentro geometria, Apollonius's theorem é um teorema relating the length of a median of a triângulo 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".

Especificamente, in any triangle

E se

is a median, então

It is a special case do Stewart's theorem. For an isosceles triangle com

the median

is perpendicular to

and the theorem reduces to the Pythagorean theorem for triangle

displaystyle ADB}

(or triangle

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.

Prova[]

The theorem can be proved as a special case of Stewart's theorem, or can be proved using vectors (Vejo 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

displaystyle a.}

Deixar

be the length of the segments of

formed by the median, assim

is half of

displaystyle a.}

Let the angles formed between

e

be

e

displaystyle theta ^{melhor },}

Onde

includes

e

displaystyle theta ^{melhor }}

includes

displaystyle c.}

Então

displaystyle theta ^{melhor }}

is the supplement of

e

displaystyle cos theta ^{melhor }=-cos theta .}

o law of cosines por

e

displaystyle theta ^{melhor }}

afirma que

Add the first and third equations to obtain

como requerido.

Veja também[]

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

Referências[]

links externos[]

• Apollonius Theorem no PlanetMath.
• David B. Surowski: Advanced High-School Mathematics. p. 27