非欧几何 Non-Euclidean Geometry Visualized.docx

Non-Euclidean Geometry VisualizedJen-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 Poincares 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 Euclideans 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 PappusStatement: If the vertices of a hexagon fall alternatively on two lines, the intersections of opposite sides are collinear.Euclidean GeometryNon-Euclidean GeometryTheorem of PascalStatement: The intersections of the opposite sides of a hexagon inscribed in a circle are collinear.Euclidean GeometryNon-Euclidean GeometryTheorem of BrianchonStatement: If a hexagon is circumscribed about a circle, the connectors of opposite vertices are concurrent.Euclidean GeometryNon-Euclidean GeometryAnother Theorem of PascalStatement: If the sides of two triangles meet in six concylic points, then they are in perspective.Euclidean GeometryNon-Euclidean GeometryTheorem of DesarguesStatement: If two triangles have a center of perspective, they have an axis of perspective.Euclidean GeometryNon-Euclidean GeometryTheorem on Doubly Perspective TrianglesStatement: Two doubly perspective triangles are in fact triply perspective.Euclidean GeometryNon-Euclidean GeometryTheorem on Triply Perspective TrianglesStatement: Two triply perspective triangles are in fact quadruply perspective.Euclidean GeometryNon-Euclidean GeometrySpecial Case of the Theorem of PappusStatement: As in the theorem of Pappus, if the vertices are in perspective, then the two given lines and the Pascal line are concurrent.Euclidean GeometryNon-Euclidean GeometryPolar of a Point with respect to a TriangleStatement: 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 GeometryNon-Euclidean GeometryProperty of a PentagonStatement: 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 GeometryNon-Euclidean GeometryConcurrent LinesLemoine PointStatement: The three symmedians of a triangle are concurrent.Euclidean GeometryNon-Euclidean GeometryOrthopoleStatement: The perpendiculars dropped upon the sides of a triangle from the projections of the opposite vertices upon a given line are concurrent.Euclidean GeometryNon-Euclidean GeometryGergonne PointStatement: The lines joining the vertices of a triangle to the points of contact of the opposite sides with the inscribed circle are concurrent.Euclidean GeometryNon-Euclidean GeometryNagal PointStatement: 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 GeometryNon-Euclidean GeometryIsotomic ConjugatesStatement: 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 GeometryNon-Euclidean GeometryIsogonal ConjugatesStatement: The isogonal conguates of the three lines joining a given point to the vertices of a given triangle are concurrent.Euclidean GeometryNon-Euclidean GeometryMittelpunktStatement: The three lines joining the excenter and the corresponding midpoint of the side of a triangle are concurrent.Euclidean GeometryNon-Euclidean GeometryPorismsSteiners PorismStatement: If two circles admit a Steiner chain, they admit an infinite number.Euclidean GeometryNon-Euclidean GeometryPoncelets PorismStatement: If two circles admit a Steiner chain, they admit an infinite number.Euclidean GeometryNon-Euclidean GeometryNone of the AboveThree-Circle TheoremStatement: The common chords of three circles taken in pairs are concurrent.Euclidean GeometryNon-Euclidean GeometryTheorem of Three Mutually Tangential CirclesStatement: The common tangents of three mutually tangential circles taken in pairs are concurrent.Euclidean GeometryNon-Euclidean GeometryTheorem of Four CirclesStatement: 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 GeometryNon-Euclidean GeometryTheorem of a Chain of Four Tangential CirclesStatement: If four circles are situated such that each touches exactly two others, then the four points of contact are concyclic.Euclidean GeometryNon-Euclidean GeometryButterfly TheoremStatement: 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 GeometryNon-Euclidean Geometry Monges TheoremStatement: The three external centers of simititudes of three circles are collinear.Euclidean GeometryNon-Euclidean GeometryReferencesI. Ya. Bakelman, InversionsN.V. Efimov, Highe