Leibniz's theorem

This article deals with Leibniz's theorem of plane geometry. For the number-theoretical theorem, possibly also proven by Leibniz for the first time, see Wilson's theorem .

Leibniz theorem for the triangle ${\displaystyle \Delta =\Delta _{ABC},}$ ${\displaystyle P=S}$
${\displaystyle a^{2}+b^{2}+c^{2}=3\cdot d^{2}+e^{2}+f^{2}+g^{2}}$

The set of Leibniz is a mathematical theorem that within the plane geometry is settled and Gottfried Wilhelm Leibniz is attributed. It gives a general formula which in particular allows the distances between the point and the corner points to be related to the distances between the corner points and the center of gravity in the Euclidean plane for a given point and a given triangle .

Formulation of the sentence

The sentence says the following:

Let four points be given in the real coordinate plane. ${\displaystyle {\mathbb {R} }^{2}}$ ${\displaystyle A,B,C,P}$

Here was the point in terms of points , the affine representation ${\displaystyle P}$ ${\displaystyle A,B,C}$

${\displaystyle P=\alpha A+\beta B+\gamma C}$with   . ${\displaystyle \alpha ,\beta ,\gamma \in \mathbb {R} \;,\;\alpha +\beta +\gamma =1}$

Let it be another arbitrary point on the real coordinate plane. ${\displaystyle X}$

Then the identity applies :

(1) ${\displaystyle \alpha \cdot |A-X|^{2}+\beta \cdot |B-X|^{2}+\gamma \cdot |C-X|^{2}-|P-X|^{2}=\alpha \cdot |A-P|^{2}+\beta \cdot |B-P|^{2}+\gamma \cdot |C-P|^{2}}$

If, in particular, the center of gravity of the triangle formed by the points is with , then it even applies ${\displaystyle P}$ ${\displaystyle A,B,C}$ ${\displaystyle \Delta =\Delta _{ABC}}$ ${\displaystyle P=S=S_{\Delta }}$ ${\displaystyle \alpha =\beta =\gamma ={\frac {1}{3}}}$

(2)   . ${\displaystyle |A-X|^{2}+|B-X|^{2}+|C-X|^{2}=3\cdot |S-X|^{2}+|A-S|^{2}+|B-S|^{2}+|C-S|^{2}}$

Note on the derivation of the sentence

The theorem allows a simple, purely computational derivation using the real scalar product by multiplying the binomial identity equation

${\displaystyle |X-Y|^{2}=|X|^{2}-2\cdot \langle X\;,\;Y\rangle +|Y|^{2}\;\;(X,Y\in {\mathbb {R} }^{n},\;n\in \mathbb {N} )}$

is applied.

Inference

The above second part of Leibniz's theorem leads directly to the following characterization of the center of gravity of a triangle, which is attributed to the Italian mathematician Giulio Carlo Fagnano and is called under the keyword Fagnanoscher's center of gravity theorem :

The center of gravity of a triangle is that point of the plane in which the sum of the squares of the distances to the three corner points ${\displaystyle \Delta =\Delta _{ABC}}$ ${\displaystyle X_{0}}$

${\displaystyle |A-X_{0}|^{2}+|B-X_{0}|^{2}+|C-X_{0}|^{2}}$

assumes the smallest value .

annotation

In Heinrich Dörries Mathematische Miniatures an analog equation is formulated about the center of gravity of a tetrahedron . In the register there, both of Dörrie's equations are referred to as Leibniz's centroid theorems.

swell

• Heinrich Dörrie: Mathematical miniatures . Second unaltered reprint of the 1943 edition. Sendet (inter alia), Wiesbaden 1979, ISBN 3-500-21150-X .
• Siegfried Gottwald , Hans-Joachim Ilgauds and Karl-Heinz Schlote (ed.): Lexicon of important mathematicians . Verlag Harri Deutsch , Thun 1990, ISBN 3-8171-1164-9 , p. 262 . MR1089881
• Max Koecher , Aloys Krieg : level geometry (=  Springer textbook ). 2nd, revised and expanded edition. Springer Verlag , Berlin (among others) 2000, ISBN 3-540-67643-0 .

References and footnotes

1. ^ Koecher, Krieg: level geometry. 2000, p. 163 and 3rd edition, 2007, p. 180
2. ↑ The proof can be found in the evidence archive .
3. ↑ Siegfried Gottwald, Hans-Joachim Ilgauds, Karl-Heinz Schlote (ed.): Lexicon of important mathematicians. 1990, p. 142
4. ^ Heinrich Dörrie: Mathematische Miniatures , 1979, pp. 273–275, p. 523