## 99 Points of Intersection: Examples-Pictures-ProofsTranslated from the original German by Peter Hilton and Jean Pedersen. The 99 points of intersection presented here were collected during a year-long search for surprising concurrence of lines. For each example we find compelling evidence for the sometimes startling fact that in a geometric figure three straight lines, or sometimes circles, pass through one and the same point. Of course, we are familiar with some examples of this from basic elementary geometry - the intersection of medians, altitudes, angle bisectors, and perpendicular bisectors of sides of a triangle. Here there are many more examples - some for figures other than triangles, some where even more than three straight lines pass through a common point. The main part of the book presents 99 points of intersection purely visually. They are developed in a sequence of figures, many without caption or verbal commentary. In addition the book contains general thoughts on and examples of the points of intersection, as well as some typical methods of proving their existence. Many of the examples shown in the book were inspired by questions and suggestions made by students and high-school teachers. Several of those examples have not only a geometrical, but also an intriguing aesthetic, aspect. The book addresses high-school students and students at the undergraduate level as well as their teachers, but will appeal to anyone interested in geometry. |

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99 Points of Intersection: Examples-Pictures-Proofs Hans Walser,Jean Pedersen No preview available - 2006 |

### Common terms and phrases

99 points afﬁne image Ai+1Ai+2 ai+2 altitudes angle bisectors angle version Author’s Cabri Geometry center of gravity centroid Ceva Ceva’s Theorem Chebyshev polynomials circle-reﬂection circles of Apollonius circumcircle common point Computer-Algebra System concurrent coordinates diagonal as chord diagonals Point dodecagon Dynamic Geometry Software English equilateral triangle Euler line Eves Mathematical example Fermat-point ﬁgure ﬁnd ﬁrst ﬂowers follows functions ghostly curves Giovanni Ceva Golden Section hexagon hyperbola incircle incircles Point intersection 16 isosceles triangles Jacobi’s Theorem Jean Pedersen l8-gon with diagonals Law of Cosines Lissajous curve Martin Gardner measures one third midpoint moving mode n-gon Optical effect original triangle parametric representation perpendicular perspectively similar Peter Hilton Petra Point of intersection Pythagoras ratios reﬂection segment W measures sheaf of straight Sheaves of circles side similar triangles sinaoz straight lines AiBi superimpose three circles three diagonals three straight lines three vertex three vertex-transversals transversals Underwood Dudley vertices Walser Wolfgang Kroll Point