Ptolemy's theorem

The set of Ptolemy (after Claudius Ptolemy ) a statement from elementary geometry that describes a relationship between the sides and diagonals of a quadrilateral tendon.

Statement

The set of Ptolemy is:

In a cyclic quadrilateral thus applies:

In addition, the reversal of the sentence of Ptolemy applies, that is true in a convex quadrilateral match the product of the diagonals with the sum of the products of the opposite sides, it is a cyclic quadrilateral. More precisely, the following statement, which is also called Ptolemy's inequality is valid:

Evidence

For a cyclic quadrilateral, consider the triangle with the separate point D on his circle of radius r and the associated Fußpunktdreieck. The formula to calculate the side lengths of a Fußpunktdreieckes then provides for:

Since D but is now on the radius, is degenerate and its sides lie on the associated Simson straight, so that the two sides LM and NM are complementary to the third page LN. Thus:

With the above equations, this provides:

If D is not on the perimeter, so due to the triangle inequality for the following applies:

The above equations then provide so that the inequality of Ptolemy:

Proof of the theorem in the complex Ptolemaic

Besides the possibility to carry out the proof elementary geometric, the Ptolemaic theorem can also be easily proved using the methods of complex analysis by looking at the properties of the complex inverse function:

Exploits.

The complex inverse function is one of the Möbius transformations, which are treated in complex analysis as a continuous Transformat ionenes erweitereten the complex plane.

(I) simplification of the problem

First, without loss of generality may assume without loss that the figure, which consists of the given cyclic quadrilateral and the associated circular line represents a geometric figure within the complex plane.

In this case, we may assume further that a special character is present with, for thus coincides with the vertex the origin. Because the theorem follows for this particular case, he generally follows, since any given geometrical figure of this type is congruent with such a special figure. Such congruence can create by means of a shift suitably chosen always.

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

Key to the proof is now the fact that for the circular line of the dotted arc under the inverse function in a straight line, namely in the image straight from below, passes over.

There lies on the dotted circular arc between the points and the point, same applies to the three pixels of the image line. So it is between and making it one of the points of the intervening distance.

(III ) Actual calculation

From (II ) follows directly using the complex absolute value function:

And thus:

And then to expand with:

And further because:

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

Generalizations ( metric spaces and Riemannian manifolds )

In CAT ( 0) - spaces, the Ptolemaic inequality holds

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

If a Riemannian manifold has non-positive sectional curvature, then it is locally Ptolemaic, ie at any point there is a neighborhood within which the Ptolemaic inequality holds.