Apollonius's theorem

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green/blue areas = red area Apollonius' theorem.svg
green/blue areas = red area
Pythagoras as a special case:
green area = red area Appolonius theorem.svg
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".

Contents

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 ApolloniusTheoremProof.svg
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

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References

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