99 Points of Intersection: Examples-Pictures-Proofs
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, developed in a sequence of figures. 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.
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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