1、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 illustratio
2、ns 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 appr
3、oach 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.
4、 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 Geo
5、metry 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 vertic
6、es 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
7、 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 Tri
8、ply 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 a
9、nd 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 m
10、eet 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 wi
11、th 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
12、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 Ge
13、ometry 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 ve
14、rtices 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 respecti
15、vely 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. Eucli
16、dean 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 c
17、hain, 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
18、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
19、 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 St
20、atement: 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.
21、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
22、 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-dimensiona
23、l 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
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