Old and New Topics in Geometry: Volume II: Advanced Euclidean and Hyperbolic Geometry

By Franz Rothe

A decade long experience of teaching the course "Fundamental of Geometry", many notes for exercises, and endless extra reading are the bases for this bulky work.

The online manuscript already includes many topics with many exercises including solutions and hundreds a elaborate computer generated drawings.

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• Language: English
• Publisher: LitPrime Solutions
• Release date: May 8, 2023
• ISBN: 9798887032535

Franz Rothe

Franz Rothe graduated from high school in Karlsruhe and studied mathematics, physics, and music there. Graduated with a diploma in mathematics from E T H Zürich, a doctorate in Tübingen. After some changes in life, a professorship at the University of North Carolina at Charlotte, USA.In addition, Rothe and pianist Thomas Turner have developed a repertoire of classical music for flute and piano, and have recorded and released three CDs. This collection also contains several of their own transcriptions. Now Rothe is retired and keeps writing books about mathematics, and too, just for entertainment.

Book preview

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Contents

I Introduction

I.1 Hilbert’s Axiomatic Approach, a Short Review

I.1.1 Logic

I.1.2 David Hilbert’s axiomatization of Euclidean geometry

I.1.3 Frege’s critique and Hilbert’s answer

I.1.4 Drawbacks and lacuna of Hilbert’s foundations

I.1.5 General remark about models in mathematics

I.2 Citations from Volume I

VI Intermediate Euclidean Geometry

VI.26 Standard Euclidean Triangle Geometry

VI.26.1 The circum-center

VI.26.2 Double and half size triangles

VI.26.3 The centroid

VI.26.4 The orthocenter

VI.26.5 The in-circle and the three ex-circles

VI.26.6 The road to the orthocenter via the orthic triangle

VI.26.7 The Euler line

VI.27 Advanced Euclidean Triangle Geometry

VI.27.1 Morley’s Theorem

VI.27.2 The Nine-Point Circle

VI.27.3 Proof of Euler’s Theorem

VI.27.4 Proof of the Nine-Point Theorem

VI.27.5 Proof of Feuerbach’s Theorem

VI.27.6 Additional questions

VI.27.7 The Simson line

VI.28 Harmonic Points

VI.28.1 The Theorems of Menelaus and Cev´

VI.28.2 The circle of Apollonius

VI.28.3 An application to electrostatics

VI.28.4 The perspective view

VI.29 Advanced Euclidean Geometry

VI.29.1 A Euclidean egg

VI.29.2 The egg built from inside

VI.29.3 An equilateral triangle on three circles

VI.29.4 A triangle construction using the sum of two sides

VI.29.5 Archimedes’ Theorem of the broken chord

VI.29.6 The Theorem of Collignon

VI.29.7 Vectors and special quadrilaterals

VI.29.8 The Theoremof Ptolemy

VI.29.9 The quadrilateral of Hjelmslev

VI.30 The Regular Pentagon

VI.30.1 The Euclidean construction with the Golden Ratio

VI.30.2 Relation between the sides of pentagon and 10-gon

VI.30.3 The construction with Hilbert tools

VI.30.4 Variants of the Euclidean construction

VI.30.5 A false pentagon

VI.31 Circles, Tangents, Power and Inversion

VI.31.1 The equipower line of two circles

VI.31.2 Common tangents of two circles

VI.31.3 Definition and construction of the inverted point

VI.31.4 The gear of Peaucollier

VI.31.5 Invariance properties of inversion

VI.32 A Glimpse at Elliptic Geometry

VI.32.1 Elliptic geometry is derived from spherical geometry

VI.32.2 The conformalmodel

VI.32.3 Falsehood of the exterior angle theorem

VI.32.4 Area of a spherical triangle

VI.32.5 Does Pythagoras’ imply the parallel postulate?

VI.32.6 The stereographic projection

VI.33 Pappus’ and Pascal’s Theorems

VI.33.1 Pappus’ Theorem in Euclidean geometry

VI.33.2 Pascal’s Theorem

VII Constructions and Their Impossibility

VII.34 Euclidean Constructions with Restricted Means

VII.34.1 Constructions by straightedge and unit measure

VII.34.2 Tools equivalent to straightedge and compass

VII.34.3 A derivation of the Theorem of Poncelet and Steiner

VII.34.4 Construction with rusty compass

VII.34.5 Hilbert tools and Euclidean tools differ in strength

VII.35 Trisection of an Angle and the Delian Problem

VII.35.1 Trisection by Archimedes

VII.35.2 Trisection by Nicomedes

VII.35.3 Trisection with Nicolson’s angle, and by origami

VII.35.4 Construction of the cubic root by two-marked ruler

VII.35.5 Definition and equations of the conchoid

VII.35.6 The conchoid and the construction