462 Index interior lemma, 97 interpretation of an axiomatic system, 26 intersecting chords theorem, 261 intersecting lines in incidence geometry, 24 in plane geometry, 55 intersecting secants theorem, 261 intersection of sets, 425 inward-pointing ray, 172 irrational number, 434 irrationality of p 2, 5 isomorphic models, 29 isomorphism between models, 29, 132, 243 isosceles right triangle, 190, 233 isosceles triangle, 104 base, 105 base angles, 105 isosceles triangle altitude theorem, 145 isosceles triangle theorem, 109 converse, 110 Pappus’s proof, 110 iterated quadratic extension, 311 Jordan polygon theorem, 167 justifications for steps in a proof, 406 Khayyam, Omar, 9, 10, 341 kite, 183, 210 Klein geometry, 373 Klein, Felix, 35, 373 Lambert quadrilateral, 341–343 Lambert, Johann Heinrich, 341 law of cosines, 240 law of sines, 241 least upper bound, 280, 435, 438 least upper bound property, 280 leg of a right triangle, 105 of a Saccheri quadrilateral, 341 Legendre, Adrien-Marie, 326 lemma, 6, 24 length of a diameter, 248 of a segment, 64 of a vector, 125 of an arc, 293 let, in mathematical proofs, 411 lies on as a primitive term, 24 in incidence geometry, 24 in plane geometry, 55 line as a primitive term, 24, 54 contains infinitely many points, 57 Euclid’s definition, 2 in incidence geometry, 24 in plane geometry, 54 in single elliptic geometry, 35, 136 in spherical geometry, 34, 135 in taxicab geometry, 138 in the Beltrami–Klein disk, 35 in the Cartesian plane, 33, 125 in the Poincar´ disk, 35, 132 in the Poincar´ half-plane, 36 in the rational plane, 136 is convex, 80 straight, 2 line segment, see segment line-circle theorem, 250 linear pair, 92 linear pair theorem, 92 converse, 93 linear polynomial, 314 linear triple, 94, 101 linear triple theorem, 94 Lobachevskian geometry, 337 Lobachevsky, Nikolai, xiii, 10 logic, laws of, 408 logical connective, 385 logical definition, 2 lower bound, 435 lowest terms, 438 major arc, 254 map, 428 mapping, 428 mathematical definition, 401 mathematical induction, 420–422 mathematical object, 24, 383, 423 mathematical relation, 384 mathematical statement, 383 mean proportional, 236 measure of an angle, see angle measure of an arc, see arc measure median of a triangle, 145 of an isosceles triangle, 145 median concurrence theorem, 224 meet in incidence geometry, 24 in plane geometry, 55 member of a set, 423

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