Index 463 membership criterion for a set, 423 Menalaus of Alexandria, 221 Menelaus’s theorem, 221 meridian circle, 35 method of exhaustion, 279 method of superposition, 15 midpoint, 68 existence and uniqueness, 69 midsegment of a Saccheri quadrilateral, 342 of a triangle, 196 midsegment theorem, 196 minor arc, 254 model of an axiomatic system, 26 monic polynomial, 314 multiple quantifiers, 399 multiplicative inverse, 434 n-gon, 157 n-point plane, 27–29 necessary condition, 389 negation, 387 negative number, 434 neutral geometry, 53 is consistent, 132 postulates of, 123, 381 nine-point circle theorem, 277 non-Euclidean geometry, 10–13 nonagon, 157 noncollinear, 25 nonconvex polygon, 166–174 nonexistence statement, 399, 420 nonnegative, 434 nonoverlapping regions, 199 nonpositive, 434 nonvertical line, 33, 125, 136 nonzero, 434 north pole, 35 not (logical negation), 387 object, mathematical, 24, 383, 423 obtuse angle, 88 obtuse triangle, 105 octagon, 157 constructing, 307 odd integer, 434 odd parity, 167 one-dimensional quantity, 226 one-point geometry, 30, 135 one-to-one correspondence, 429 one-to-one function, 429 one-two geometry, 31 onto, 429 open half-plane, 77 open interval, 434 open sentence, 394 opposite angle of a quadrilateral, 175 of a triangle, 104 opposite ray theorem, 75 opposite rays, 74, 75 opposite side of a line, 76 of a quadrilateral, 175 of a triangle, 104 opposite vertices of a quadrilateral, 175 or (logical disjunction), 386 ordered n-tuple, 426 ordered pair, 426 equality of, 426 ordering lemma for points, 73 for rays, 98 origin in the Cartesian plane, 124 orthocenter, 269 outward-pointing ray, 172 overlapping regions, 199 , definition of, 289 P -side of an edge line, 158 pair, ordered, 426 Pappus of Alexandria, 110 parallel lines are equidistant, 187 constructing, 300 existence of, 153 in incidence geometry, 25 in plane geometry, 55, 149–154 parallel postulate elliptic, 38 Euclidean, 38, 185 hyperbolic, 38, 337 in single elliptic geometry, 41 in spherical geometry, 41 in the Beltrami–Klein disk, 41 in the Cartesian plane, 40 in the hyperboloid model, 41 in the Poincar´ disk, 41 in the Poincar´ half-plane, 41 independence, in incidence geometry, 39 independence, in neutral geometry, 133 parallel projection theorem, 220 parallelism, transitivity of, 188, 323 parallelogram, 175, 194

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