Beitrag zum Jahr der Mathematik 2008

Table of Contents

Einleitung

Erster TEIL:

Wehrle-Formeln für Dreiecke

Rationale Dreiecke mit natürlichen Seitenlängen

Abstand der Zentren

Der Sinus-Wehrle

Andere trigonometrische Wehrles

Des Sinus-Differenzen-Wehrle

Die trigonomischen Potenzen-Wehrles

Die Summe der Seitenquadrate des Dreiecks

Die Kreise des Dreiecks

Das Küßproblem

Die Formel von Descartes

Die Kreisspiegelung

Die Krümmung der Küßkreise

Die Krümmung der Ankreise

Die Eulergerade und der Feuerbachkreis

Merkwürdigkeiten beim Dreieck

Teil II: Das Radienprodukt für Kreisvierecke

Flächeninhalt

Sehnen- und Tangentenvierecke

Der Drachen-Wehrle

Rationale Vierecke mit natürlichen Seiten

Das Radienprodukt für Kreistrapeze ist

Teil III: Pyramiden

Das Analogon zum rechtwinkligen Dreieck: Der dreidimensionale Pythagoras

Die Entfernung der Zentren

Die Fehringer-Wehrle-Formel

TEIL IV: n- und unendliche Dimensionalität

Wir verlassen nun die uns vertrauten Sphären!

Satz des Pythagoras im n-Dimensionalen

Das allereinfachste Gebilde, genannt SIMPLEX

Volumen der unendlich-dimensionalen Kugeln

Schlußfolgerung

ANHANG

BEWEIS der WEHRLE-Formel

Beweis der Flächenformel

Beweis für w = R² - d²

Übungsaufgaben zum Kapitel I:

Übungsaufgaben zum Kapitel II:

Übungsaufgaben zum Kapitel III:

Objectives and Topics

This work aims to explore and derive new geometric relationships in triangles, quadrilaterals, pyramids, and n-dimensional structures, introducing novel concepts such as the "Wehrle-number" to unify various geometric properties. The primary research goal is to demonstrate how these algebraic identities hold across different dimensions and complex geometric configurations.

• Derivation of the "Wehrle-formulas" for triangle side and radius relations.
• Exploration of rational triangles and quadrilaterals with natural side lengths.
• Analysis of geometric circle properties including kissing circles, Feuerbach circles, and inversion.
• Extension of the Pythagorean theorem to 3D pyramids and n-dimensional simplices.
• Examination of hyperspheres and the behavior of volume/surface area in infinite dimensions.

Excerpt from the Book

Summary of Chapters

Erster TEIL: This section introduces the foundational "Wehrle-formulas" and investigates the relationships between side lengths, areas, and radii in both general and rational triangles.

Die Kreise des Dreiecks: This chapter focuses on advanced circle geometry, specifically the "kissing problem," circle inversion, and the properties of the Euler line and Feuerbach circle.

Teil II: Das Radienprodukt für Kreisvierecke: This part shifts focus to quadrilaterals, analyzing the conditions for cyclic and tangential quadrilaterals and extending the Wehrle concepts to these shapes.

Teil III: Pyramiden: This chapter generalizes the Pythagorean theorem to three dimensions by examining rechtwinklige Pyramiden and the specific metrics related to their inspheres and circumspheres.

TEIL IV: n- und unendliche Dimensionalität: The final chapter takes the concepts into n-dimensional and infinite-dimensional spaces, exploring the disappearance of hyperspheres and the limitations of Euclidean geometry in extreme dimensions.

Keywords

Wehrle-Zahl, Dreiecksgeometrie, Inkreis, Umkreis, Kreisviereck, rechtwinklige Pyramide, dreidimensionaler Pythagoras, n-dimensionales Simplex, Kreisspiegelung, Feuerbachkreis, Kugelradien, Geometrie, Mathematik, Radienprodukt

Frequently Asked Questions

What is the core focus of this publication?

The work focuses on developing algebraic identities in geometry, specifically linking side lengths, surface areas, and radii in triangles and higher-dimensional polyhedra using the author's defined "Wehrle-number."

What are the primary thematic areas covered?

The book spans classical triangle geometry, circle-based problems like the "kissing problem," quadrilateral geometry, and higher-dimensional geometric analysis.

What is the primary objective of the author?

The author aims to establish a consistent set of formulas (Wehrle-formulas) that describe the interplay between inscribed and circumscribed spheres/circles in varying geometric figures.

Which scientific methodology is utilized?

The author uses analytical geometry, vector analysis, and algebraic derivations based on established Euclidean principles, often verified through modern 3D visualizations.

What topics are discussed in the main body?

The body covers the properties of triangle centers, circle inversion techniques, the generalization of the Pythagorean theorem to 3D and n-dimensions, and the analysis of limits in infinite dimensions.

Which keywords define this work?

The work is defined by terms such as Wehrle-number, circle geometry, polyhedra radii, inversion, and high-dimensional geometric stability.

How is the "Wehrle-number" defined for triangles?

It is defined as the product of the side lengths divided by the perimeter, which equals twice the product of the inradius and circumradius.

What does the book conclude about Euclidean geometry in infinite dimensions?

The author concludes that Euclidean geometry reaches its limitations in infinite-dimensional spaces, suggesting it functions as an ideal model that cannot fully capture the complexity of the actual spatial limit.