Apollonios equation

This article deals with a theorem of Apollonios. For further theorems of Apollonios see Theorem of Apollonios .

The Apollonios equation is a mathematical theorem , which belongs to both the mathematical subfield of geometry and that of functional analysis . It is attributed to the ancient Greek mathematician Apollonios von Perge and deals with a basic metric relationship between sides and bisectors of triangles . The equation is closely related to the Pythagorean equation . In elementary geometry one speaks in connection with this equation also of the theorem of the bisector or the theorem of Apollonios . A well-known theorem by Leonhard Euler is a direct consequence of the Apollonios equation .

formulation

The Apollonios equation can be formulated as follows:

For three points of an interior product space , which is provided with the standard resulting from the interior product of this space , the following always applies: ${\displaystyle x,y,z}$ ${\displaystyle V}$ ${\displaystyle {\|\cdot \|}}$

(AG-1)     . ${\displaystyle {\tfrac {1}{2}}\|x-y\|^{2}+2\,\|z-{\tfrac {1}{2}}(x+y)\|^{2}=\|z-x\|^{2}+\|z-y\|^{2}}$

Explanations, comments and conclusions

• The Apollonios equation is a direct consequence of the parallelogram equation , which in turn results directly - namely for - from the Apollonios equation. ${\displaystyle z=0}$
• Here is the Euclidean plane , provided with the Euclidean norm , and is a triangle before, for which - as usual - the side lengths with and the length of the points associated medians with are named, then writes (AG-1) in the form ${\displaystyle V}$ ${\displaystyle ABC}$ ${\displaystyle a,b,c}$ ${\displaystyle C}$ ${\displaystyle s_{c}}$

(AG-2a)       , ${\displaystyle {\tfrac {1}{2}}c^{2}+2{s_{c}}^{2}=a^{2}+b^{2}}$

with which the well-known (equivalent!) formula

(AG-2b)     ${\displaystyle s_{c}={\frac {\sqrt {2(a^{2}+b^{2})-c^{2}}}{2}}}$

results.

• If one forms the corresponding formulas for the two other sides of the triangle and their bisectors, one obtains - after equivalence transformations - the three equations

(AG-3a)     ${\displaystyle {s_{a}}^{2}-{s_{b}}^{2}={\frac {3}{4}}(b^{2}-a^{2})}$

(AG-3b)     ${\displaystyle {s_{b}}^{2}-{s_{c}}^{2}={\frac {3}{4}}(c^{2}-b^{2})}$

(AG-3c)     ${\displaystyle {s_{c}}^{2}-{s_{a}}^{2}={\frac {3}{4}}(a^{2}-c^{2})}$

• It follows immediately:

(AG-F1)     ${\displaystyle {s_{a}}^{2}+{s_{b}}^{2}+{s_{c}}^{2}={\frac {3}{4}}(a^{2}+b^{2}+c^{2})}$

(AG-F2)     ${\displaystyle s_{a}\geq s_{b}\geq s_{c}{\text{ wenn }}a\leq b\leq c}$

(AG-F3)     If, in particular, a right-angled triangle is the Euclidean plane and the length of the hypotenuse , then according to the Thales theorem and thus the Pythagorean equation applies . ${\displaystyle ABC}$ ${\displaystyle c}$ ${\displaystyle s_{c}={\tfrac {1}{2}}c}$

See also

• Apollonios' theorem (triangle)
• Pythagorean theorem
• Euler's theorem (quadrangular geometry)

literature

• Ilka Agricola , Thomas Friedrich : Elementary Geometry . Expertise for studies and mathematics lessons (=  studies ). 4th, revised edition. Springer Spectrum , Wiesbaden 2015, ISBN 978-3-658-06730-4 , doi : 10.1007 / 978-3-658-06731-1 .
• Claudi Alsina - Roger B. Nelsen : Pearls of Mathematics . 20 geometric figures as starting points for mathematical explorations. Springer Spectrum , Berlin - Heidelberg 2015, ISBN 978-3-662-45461-9 .
• IN Bronstein , KA Semendjajew : Pocket book of mathematics . 12th edition. Harri Deutsch publishing house , Frankfurt am Main 1972.
• Jürgen Heine : Topology and Functional Analysis . Basics of abstract analysis with applications. 2nd, improved edition. Oldenbourg Verlag , Munich 2011, ISBN 978-3-486-70530-0 .

Footnotes and individual references

1. ↑ ^{a b} J. Heine: Topology and Functional Analysis. 2002, p. 25
2. ^ Ilka Agricola, Thomas Friedrich: Elementarge Geometry. 2015, p. 77
3. ↑ ^{a b c} Claudi Alsina, Roger B. Nelsen: Pearls of Mathematics. 2015, p. 63
4. ↑ J. Heine, op.cit., P. 551
5. ↑ By solving the equation (AG-2a) according to . ${\displaystyle s_{c}}$
6. ^ Bronstein / Semendjajew 1972, p. 141