Apollonios' theorem (triangle)

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(blue-green and) green area = red area

The theorem of Apollonius , named after Apollonios von Perge (265 BC - 190 BC), describes an equality of area in any triangle with an associated side bisector .

Let ABC be a triangle and AD be the bisector of side BC , then the following equation applies:

The theorem also results as a special case of Stewart's theorem, which gives an equation for the case of an arbitrary Cevane instead of a bisector. In contrast to Apollonios' theorem, the sizes occurring in Stewart's theorem cannot be directly interpreted as areas.

Pythagoras as a special case:
green area = red area

If the triangle ABC has a right angle in A , then the Pythagorean theorem is obtained from the Apollonios theorem as a corollary , since the bisector AD then corresponds to the radius of the corresponding Thales circle . That means that:

This then results in the Pythagorean theorem:

If one doubles the triangle ABC to a parallelogram, so that the side BC becomes one of the diagonals of the parallelogram, then Apollonios' theorem gives the parallelogram equation . Outside of elementary geometry, one speaks accordingly of the Apollonios equation .

The reverse of Apollonios' theorem also applies, that is, if the above equation is satisfied for any point D on BC , then D is the center of BC .

literature

  • Claudi Alsina, Roger B. Nelsen: Pearls of Mathematics: 20 geometric figures as starting points for mathematical exploratory trips . Springer, 2015, ISBN 978-3-662-45461-9 , p. 63 ( excerpt from the English edition books.google.de)
  • AJG May: 1535. The Converse of Apollonius' Theorem . In: The Mathematical Gazette , Volume 25, No. 266 (Oct. 1941), pp. 228-229 ( JSTOR 3606591 )

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