En géométrie, 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".
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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
is a median, alors
It is a special case de Stewart's theorem. For an isosceles triangle avec
is perpendicular to
and the theorem reduces to the Pythagorean theorem for 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.
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.
Let the triangle have sides
with a median
drawn to side
be the length of the segments of
formed by the median, alors
is half of
Let the angles formed between
is the supplement of
La law of cosines pour
Add the first and third equations to obtain
- Formulas involving the medians' lengths – Line segment joining a triangle's vertex to the midpoint of the opposite side
Godfrey, Charles; Siddons, Arthur Warry (1908). Modern Geometry. University Press. p.20.
- Apollonius Theorem à PlanèteMath.
- David B. Surowski: Advanced High-School Mathematics. p. 27
Si vous voulez connaître d'autres articles similaires à Apollonius's theorem vous pouvez visiter la catégorie Géométrie euclidienne.
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