Dans géométrie, Apollonius's theorem est un théorème 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".

Spécifiquement, in any triangle

si

is a median, alors

It is a special case de Stewart's theorem. For an isosceles triangle avec

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.

Preuve[]

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

Laisser

be the length of the segments of

formed by the median, alors

is half of

displaystyle a.}

Let the angles formed between

et

be

et

displaystyle theta ^{prime },}

où

includes

et

displaystyle theta ^{prime }}

includes

displaystyle c.}

Alors

displaystyle theta ^{prime }}

is the supplement of

et

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

La law of cosines pour

et

displaystyle theta ^{prime }}

stipule que

Add the first and third equations to obtain

comme demandé.

Voir également[]

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

Références[]

Liens externes[]

• Apollonius Theorem à PlanèteMath.
• David B. Surowski: Advanced High-School Mathematics. p. 27