464 Index area of, 209 congruent opposite angles, 180, 194 congruent opposite sides, 181, 194 diagonals bisect each other, 182, 194 is convex, 177 properties, 194 parallelogram decomposition lemma, 203 parallelogram lemma, 178 parity of a point, 167 of a ray, 167 Pascal’s mystic hexagon, 277 Pasch’s axiom, 105 Pasch’s theorem, 105 for asymptotic triangles, 353 Pasch, Moritz, 105 passing point, 65 strong, 174 path, polygonal, 167 pentagon, 157 constructing, 306 perimeter of a polygon, 226 perimeter scaling theorem, 226 perpendicular bisector, 145 constructing, 299 perpendicular bisector concurrence theorem, 264 perpendicular bisector theorem, 145 converse, 146 perpendicular lines, 101, 141 constructing, 101, 299, 300 dropping, 142, 300 Euclid’s definition, 2 perpendicular rays, 141 perpendicular segments, 141 pi, definition of, 289 pizza lemma, 202 plane geometry, 53 plane separation postulate, 76 plane, the, 55 Playfair’s postulate, 10, 323 Playfair, John, 10 Poincar´ disk, 35–36, 41, 132–135 is a model of hyperbolic geometry, 133 is a model of incidence geometry, 36 is a model of neutral geometry, 133 Poincar´ half-plane, 36, 41 is a model of incidence geometry, 36 point as a primitive term, 24, 54 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, 36 in the Cartesian plane, 33, 125 in the Poincar´ disk, 35, 132 in the Poincar´ half-plane, 36 in the rational plane, 136 point of tangency, 249 polygon, 155 area of, 200 concave, 158 convex, 157–166, 173 interior of, 164, 171 nonconvex, 166–174 regular, 159, 271–273 polygon splitting theorem, 163 polygonal path, 167 polygonal region, 200 general, 200 simple, 199 polynomial, 314 positive number, 434 postulate, 13, 23 angle measure, 85 area, 209 distance, 56 elliptic parallel, 38 Euclidean area, 200 Euclidean parallel, 38, 185 existence, 55 hyperbolic parallel, 38, 337 of Euclid, 3 of Euclidean geometry, 123, 382 of hyperbolic geometry, 123, 382 of incidence geometry, 25 of neutral geometry, 123, 381 plane separation, 76 protractor, 85 reflection, 443 ruler, 57 SAS, 108 set, 55 unique line, 56 predicate, 394 premise of a rule of inference, 408 prime number, 434 primitive term, 23, 54 Proclus, 6, 9, 186

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