Ptolemy's theorem

Tendon quadrangle

Ptolemy's inequality

The set of Ptolemäus (after Claudius Ptolemäus ) is a theorem of elementary geometry , a relationship between the sides and diagonals of a tendon quadrilateral describes. It can be understood as a generalization of the Pythagorean theorem and is itself a borderline case of Casey's theorem .

statement

Ptolemy's theorem reads:

In a chordal quadrilateral, the product of the lengths of the diagonals is equal to the sum of the products of the lengths of opposite sides .

In a chordal quadrilateral the following applies ${\displaystyle \square ABCD}$

${\displaystyle |AC|\cdot |BD|=|AB|\cdot |CD|+|BC|\cdot |AD|}$

In addition, the reverse of Ptolemy's theorem also applies, i.e. if the product of the diagonals in a convex square corresponds to the sum of the products of the opposite sides, then it is a chordal quadrilateral. The following statement, which is also known as Ptolemy's inequality, applies a little more precisely :

Let be a triangle and ${\displaystyle \triangle ABC}$ D a point that does not lie on the arc of the circumference, then: ${\displaystyle {\widehat {AC}}}$

${\displaystyle |AB|\cdot |CD|+|BC|\cdot |AD|>|AC|\cdot |BD|}$

proof

Ptolemy's inequality

In the case of a chordal quadrilateral , consider the triangle with the separate point D on its circumference with radius r and the corresponding base point triangle . The formula for calculating the side lengths of a base triangle then provides for : ${\displaystyle \square ABCD}$ ${\displaystyle \triangle ABC}$ ${\displaystyle \triangle LMN}$ ${\displaystyle \triangle LMN}$

${\displaystyle {\begin{aligned}&|MN|&={\frac {|AD|\cdot |BC|}{2r}}\\&|LN|&={\frac {|BD|\cdot |AC|}{2r}}\\&|LM|&={\frac {|CD|\cdot |AB|}{2r}}\\\end{aligned}}}$

Ptolemy's inequality

Since D now lies on the circumference of , is degenerate and its sides lie on the associated Simson straight line , so that the two sides LM and NM complement each other to form the third side LN . The following applies: ${\displaystyle \triangle ABC}$ ${\displaystyle \triangle LMN}$

${\displaystyle |LM|+|NM|=|LN|}$

With the above equations this gives:

${\displaystyle |AB|\cdot |CD|+|BC|\cdot |AD|=|AC|\cdot |BD|}$

If D is not on the circumference, then due to the triangle inequality for : ${\displaystyle \triangle LMN}$

${\displaystyle |LM|+|NM|>|LN|}$

The above equations then yield Ptolemy's inequality:

${\displaystyle |AB|\cdot |CD|+|BC|\cdot |AD|>|AC|\cdot |BD|}$

Proof of the Ptolemaic theorem in the complex

In addition to the possibility of proving the proof in terms of elementary geometry, the Ptolemaic theorem can also be easily proven with methods of complex analysis by considering the properties of the complex inverse function :

${\displaystyle I:\;z\mapsto I(z)={\frac {1}{z}}}$

exploits.

The complex inverse function is one of the Möbius transformations , which are treated in complex analysis as continuous transformations of the extended complex number level.

(I) simplifying the problem

First of all, one can assume that the figure, which consists of the given chordal quadrangle and the associated circular line , represents a geometric figure within the complex plane of numbers. ${\displaystyle \square ABCD}$ ${\displaystyle k}$ ${\displaystyle \mathbb {C} }$

Here, one may assume further that a special figure with present, for that is the cornerstone to the origin coincides. For if the proposition for this special case follows, then it follows generally, since every given geometrical figure of the kind mentioned is congruent to such a special figure. Such a congruence can always be created by means of a suitably selected shift . ${\displaystyle A=0}$ ${\displaystyle A}$ ${\displaystyle 0\in \mathbb {C} }$

(II) Use of the geometric properties of the inverse function

What is essential for the proof is the fact that for the circular line, the dotted circular arc merges into a straight line under the inverse function , namely into the image straight line from below . ${\displaystyle k}$ ${\displaystyle k\setminus \{0\}}$ ${\displaystyle I}$ ${\displaystyle g=\{I(z):{z\in k}\wedge {z\neq 0}\}}$ ${\displaystyle k\setminus \{0\}}$ ${\displaystyle I}$

Since the point now lies between the points and on the dotted circular arc, the same applies to the three image points of the straight line. So it lies between and and thus belongs to the points of the line in between. ${\displaystyle C}$ ${\displaystyle B}$ ${\displaystyle D}$ ${\displaystyle I(C)={\frac {1}{C}}}$ ${\displaystyle I(B)={\frac {1}{B}}}$ ${\displaystyle I(D)={\frac {1}{D}}}$

(III) Actual calculation

Using the complex amount function, (II) immediately results in : ${\displaystyle z\mapsto |z|={\sqrt {z\cdot {\overline {z}}}}}$

${\displaystyle \left|{\frac {1}{D}}-{\frac {1}{B}}\right|=\left|{\frac {1}{D}}-{\frac {1}{C}}\right|+\left|{\frac {1}{C}}-{\frac {1}{B}}\right|}$

and thus:

${\displaystyle {\frac {|B-D|}{|B|\cdot |D|}}={\frac {|C-D|}{|C|\cdot |D|}}+{\frac {|B-C|}{|B|\cdot |C|}}}$

and then after expanding with  : ${\displaystyle |B|\cdot |C|\cdot |D|}$

${\displaystyle |B-D|\cdot |C|=|C-D|\cdot |B|+|B-C|\cdot |D|}$

and further because of : ${\displaystyle A=0}$

${\displaystyle |A-C|\cdot |B-D|=|A-B|\cdot |C-D|+|B-C|\cdot |A-D|}$

But this is nothing more than the identity asserted above and to be proven .

Conclusion: The Pythagorean theorem

Each rectangle is a chordal quadrilateral in which - with the above designations! - the equations hold. Since a right-angled triangle can always be supplemented to a rectangle in such a way that the hypotenuse of the right-angled triangle coincides with one of the two rectangular diagonals and the two cathets with two adjacent rectangle sides , Ptolemy's theorem entails the Pythagorean theorem. ${\displaystyle {\overline {AB}}={\overline {CD}},{\overline {BC}}={\overline {AD}},{\overline {AC}}={\overline {BD}}}$

Generalizations (metric spaces and Riemannian manifolds)

The Ptolemaic inequality applies in CAT (0) spaces ${\displaystyle (X,d)}$

${\displaystyle d(x,y)d(z,p)\leq d(x,z)d(p,y)+d(x,p)d(y,z)}$for everyone . ${\displaystyle x,y,z,p\in X}$

The converse also applies to complete Riemannian manifolds : if the Ptolemaic inequality holds for all points, then it is a CAT (0) -space.

If a Riemannian manifold has non-positive section curvature , then it is locally Ptolemaic; H. for every point there is an environment within which the Ptolemaic inequality holds.

literature

• John Roe : Elementary Geometry (=  Oxford science publications ). Oxford University Press, Oxford [u. a.] 1993, ISBN 0-19-853457-4 .
• Anna Maria Fraedrich: The sentence group of the Pythagoras (=  textbooks and monographs on didactics of mathematics . Volume 29 ). BI-Wissenschaftsverlag, Mannheim / Leipzig / Vienna / Zurich 1994, ISBN 3-411-17321-1 .
• Klaus Gürlebeck, Klaus Habetha , Wolfgang Sprößig: Function theory in the plane and in space . Birkhäuser Verlag, Basel [u. a.] 2006, ISBN 978-3-7643-7369-6 .
• Helmut Karzel , Hans-Joachim Kroll: History of Geometry since Hilbert . Scientific Book Society, Darmstadt 1988, ISBN 3-534-08524-8 .
• Hugo Fenkner, Karl Holzmüller: Mathematical teaching work. According to the guidelines for the curricula of the higher schools in Prussia, revised by Dr. Karl Holzmüller. Geometry. Edition A in 2 parts. I. part . 12th edition. Published by Otto Salle, Berlin 1926.
• Theophil Lambacher , Wilhelm Schweizer (Ed.): Lambacher-Schweizer . Mathematical teaching material for higher schools. Geometry. Edition E. Part 2 . 13th edition. Ernst Klett Verlag, Stuttgart 1965.

Individual evidence

1. ↑ ^{a b c d} H. SM Coxeter , SL Greitzer: Geometry Revisited . Math. Assoc. Amer., Washington DC 1967, pp. 23, 41–42 ( excerpt (Google) )
2. ↑ Hugo Fenkner, Karl Holz Müller: Mathematical Unterrichtswerk. According to the guidelines for the curricula of the higher schools in Prussia, revised by Dr. Karl Holzmüller. Geometry. Edition A in 2 parts. I. part . 12th edition. Verlag von Otto Salle, Berlin 1926, p. 170 .
3. ^ Theophil Lambacher , Wilhelm Schweizer (ed.): Lambacher-Schweizer . Mathematical teaching material for higher schools. Geometry. Edition E. Part 2 . 13th edition. Ernst Klett Verlag, Stuttgart 1965, p. 156 .
4. ^ John Roe : Elementary Geometry (=  Oxford science publications ). Oxford University Press, Oxford [u. a.] 1993, ISBN 0-19-853457-4 , pp. 123 ( excerpt (Google) ).
5. ↑ Helmut Karzel , Hans-Joachim Kroll: History of geometry since Hilbert . Scientific Book Society, Darmstadt 1988, ISBN 3-534-08524-8 , pp. 96 .
6. ^ Anna Maria Fraedrich: The sentence group of Pythagoras (=  textbooks and monographs on didactics of mathematics . Volume 29 ). BI-Wissenschaftsverlag, Mannheim / Leipzig / Vienna / Zurich 1994, ISBN 3-411-17321-1 , p. 63-64 .
7. ↑ SM Buckley, K. Falk, DJ Wraith: Ptolemaic Spaces and CAT (0) . (PDF; 181 kB) In: Glasg. Math. J. , 51, 2009, no. 2, pp. 301-314.
8. ↑ DC Kay: Ptolemaic metric spaces and the characterization of geodesics by vanishing metric curvature . Ph.D. thesis, Michigan State Univ., East Lansing MI 1963