Non-Euclidean Geometry Visualized

Jen-chung Chuan
Department of Mathematics
National Tsing Hua University
Hsinchu, Taiwan 300

Introduction

Browsing through publications on non-Euclidean geometry, it is clear that very few address directly the concrete theorems and very few illustrations are given. We attempt to remove the mystery by supplying here a collection of interesting theorems of non-Euclidean geometry under the Poincare's model that can be visualized.

Converting Straight Lines into Circles

Where to find the source of inspiration in non-Euclidean geometry? One approach is to examine painstakingly all theorems in the ordinary plane geometry that do not involve the Euclidean's 5th postulate. In reality a large quantities of such theorems exist in projective geometry. In projective geometry the only basic geometric objects involved are straight lines and points. Hence all theorems in projective geometry are readily converted into theorems in non-Euclidean geometry. Examples:

Theorem of Pappus
Statement: If the vertices of a hexagon fall alternatively on two lines, the intersections of opposite sides are collinear.
Euclidean Geometry	Non-Euclidean Geometry

Theorem of Pascal
Statement: The intersections of the opposite sides of a hexagon inscribed in a circle are collinear.
Euclidean Geometry	Non-Euclidean Geometry

Theorem of Brianchon
Statement: If a hexagon is circumscribed about a circle, the connectors of opposite vertices are concurrent.
Euclidean Geometry	Non-Euclidean Geometry

Another Theorem of Pascal
Statement: If the sides of two triangles meet in six concylic points, then they are in perspective.
Euclidean Geometry	Non-Euclidean Geometry

Theorem of Desargues
Statement: If two triangles have a center of perspective, they have an axis of perspective.
Euclidean Geometry	Non-Euclidean Geometry

Theorem on Doubly Perspective Triangles
Statement: Two doubly perspective triangles are in fact triply perspective.
Euclidean Geometry	Non-Euclidean Geometry

Theorem on Triply Perspective Triangles
Statement: Two triply perspective triangles are in fact quadruply perspective.
Euclidean Geometry	Non-Euclidean Geometry

Special Case of the Theorem of Pappus
Statement: As in the theorem of Pappus, if the vertices are in perspective, then the two given lines and the Pascal line are concurrent..
Euclidean Geometry	Non-Euclidean Geometry

Polar of a Point with respect to a Triangle
Statement: Let ABC be a given triangle and O an arbitrary point of the plane. Draw AO, BO, CO to meet BC, CA, AB in L, M, N respectively, and then draw MN, NL, LM to meet BC, CA, AB in U,V,W respectively. Then U,V,W are collinear.
Euclidean Geometry	Non-Euclidean Geometry

Property of a Pentagon
Statement: Let ABCDE be an arbitrary pentagon, F the point of intersection of the nonadjacent sides AB and CD, M the point of intersection of the diagonal AD with the line EF. Then the point of interestion P with the side AE with the line BM, the point of intersection Q of the side DE with the line CM, and the point of intersection R of the side BC with the diagonal AD all lie on one line.
Euclidean Geometry	Non-Euclidean Geometry

Concurrent Lines

Lemoine Point
Statement: The three symmedians of a triangle are concurrent.
Euclidean Geometry	Non-Euclidean Geometry

Orthopole
Statement: The perpendiculars dropped upon the sides of a triangle from the projections of the opposite vertices upon a given line are concurrent.
Euclidean Geometry	Non-Euclidean Geometry

Gergonne Point
Statement: The lines joining the vertices of a triangle to the points of contact of the opposite sides with the inscribed circle are concurrent.
Euclidean Geometry	Non-Euclidean Geometry

Nagal Point
Statement: The lines joining the vertices of a triangle to the points of contact of the opposite sides with the excircles relative to those sides are concurrent.
Euclidean Geometry	Non-Euclidean Geometry

Isotomic Conjugates
Statement: If the three lines joining three points marked on the sides of a triangle to the respectively opposite vertices are concurrent, the same is true of the isotomics of the given points.
Euclidean Geometry	Non-Euclidean Geometry

Isogonal Conjugates
Statement: The isogonal conguates of the three lines joining a given point to the vertices of a given triangle are concurrent.
Euclidean Geometry	Non-Euclidean Geometry

Mittelpunkt
Statement: The three lines joining the excenter and the corresponding midpoint of the side of a triangle are concurrent..
Euclidean Geometry	Non-Euclidean Geometry

Porisms

Steiner's Porism
Statement: If two circles admit a Steiner chain, they admit an infinite number.
Euclidean Geometry	Non-Euclidean Geometry

Poncelet's Porism
Statement: If two circles admit a Steiner chain, they admit an infinite number.
Euclidean Geometry	Non-Euclidean Geometry

None of the Above

Three-Circle Theorem
Statement: The common chords of three circles taken in pairs are concurrent.
Euclidean Geometry	Non-Euclidean Geometry

Theorem of Three Mutually Tangential Circles
Statement: The common tangents of three mutually tangential circles taken in pairs are concurrent.
Euclidean Geometry	Non-Euclidean Geometry

Theorem of Four Circles
Statement: Given four concyclic points A,B,C,D, if four circles through AB, BC,CD,DA are drawn, then the remaining four intersections of succesive circles are concyclic.
Euclidean Geometry	Non-Euclidean Geometry

Theorem of a Chain of Four Tangential Circles
Statement: If four circles are situated such that each touches exactly two others, then the four points of contact are concyclic.
Euclidean Geometry	Non-Euclidean Geometry

Butterfly Theorem
Statement: Given a chord PQ of a circle, draw any two chords AB and CD passing through its midpoint. Call the points where AD and BC meet PQ X and Y. Then M is the midpoint of XY.
Euclidean Geometry	Non-Euclidean Geometry

Monge's Theorem
Statement: The three external centers of simititudes of three circles are collinear.
Euclidean Geometry	Non-Euclidean Geometry

References

I. Ya. Bakel'man, Inversions
N.V. Efimov, Higher Geometry
Howard Eves, Whitley, A survey of geometry
H.G. Forder, Geometry, An Introduction
E. A. Maxwell, Geometry for Advanced Pupils
C. Stanley Ogilvy, Excursions in geometry
Hans Schwerdtfeger, Geometry of Complex Numbers; III two-dimensional Non-Euclidean Geometries
I. M. Yaglom, A simple non-Euclidean geometry and its physical basis: an elementary account of Galilean geometry and the Galilean principle of relativity
I.M. Yaglom, Geometric Transformations I
I.M. Yaglom, Geometric Transformations II
I.M. Yaglom, Geometric Transformations III
I.M. Yaglom, Complex Numbers in Geometry; Appendix: Non-Euclidean Geometries in the plane and Complex Numbers