Index 457 Beltrami, Eugenio, 11, 35 Beltrami–Klein disk, 35, 41 betweenness consistency of, 61, 90 in Hilbert’s axioms, 20, 23, 375 of numbers, 59 of points, 60 of rays, 89 symmetry of, 60, 89 vs. betweenness, 99 vs. interior, 97 betweenness axiom, Hilbert’s, 61, 375 betweenness theorem converse, 61, 117 for points, 60 for rays, 89 biconditional statement, 392 bijective function, 57, 429 Birkhoff’s axioms, 377 Birkhoff, George D., 20, 53, 377 bisect, 68 bisector of an angle, 100, 147, 298 perpendicular, 145, 146 Bolyai, Janos, ´ xiii, 10 boundary of a simple polygonal region, 199 braces in set notation, 424 Cartesian plane, 33–34, 125–132 is a model of Euclidean geometry, 132 is a model of incidence geometry, 33 is a model of neutral geometry, 132 Cartesian product, 426 cases, proof by, 414 categorical axiomatic system, 132, 243 Euclidean geometry, 132, 243 center of a circle, 70, 247 of a regular polygon, 271 central angle of a circle, 257 of a regular polygon, 271 centroid, 224 Ceva’s theorem, 223 Ceva, Giovanni, 223 cevian, 223 chord of a circle, 35, 247, 250 of a polygon, 161, 162 circle, 70, 247 area of, 293 circumference of, 289 through three noncollinear points, 264 unit, 289 circular region, 290 area of, 291 circumcenter, 262 circumcircle, 262 for a triangle, 263 circumcircle postulate, 335 circumcircle theorem, 263 circumference, 284 approximation by regular polygons, 285 formula for, 289 scaling theorem, 288 circumscribed circle, 262 constructing, 305 for a triangle, 263 circumscribed polygon, 265 is convex, 266 regular, 273 Clairaut’s postulate, 330 Clairaut, Alexis, 330 classification of parallels through a point, 357 closed half-plane, 77 closed interval, 434 closest point, 146 on a line, 147 on a segment, 147 closure properties of numbers, 435 codomain of a function, 428 coefficient of a polynomial, 314 collection (synonym for set), 423 collinear points in incidence geometry, 25 in plane geometry, 55 collinear rays, 74 common notions for angles, 90 for segments, 66 of Euclid, 4 common perpendicular, 151, 187, 359, 360 uniqueness, 359 common perpendicular theorem, 151 converse, 187 compass and straightedge, 3, 295, 296 complementary angles, 91 components of a vector, 124 of an ordered pair, 426 composition of rigid motions, 449 compound statement, 385

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