460 Index equiangular quadrilateral, 181, 182, 194 equiangular triangle, 105, 110 equidistance postulate, 323 equidistance theorem, 152 converse, 187 equidistant from a line, 151 from two lines, 147 from two points, 68 equidistant lines, 151, 152, 187, 362 symmetry of, 188 equilateral polygon, 159 equilateral triangle, 104, 110 construction theorem, 191, 296 equivalence, 392 proof of, 413 equivalence relation, 393 equivalent postulates, 191, 321 Erchinger, Johannes, 318 Erlangen program, 373 Euclid, 1 Euclid’s Elements, 1 Euclid’s Postulate 1, 3 Euclid’s Postulate 2, 3 Euclid’s Postulate 3, 3, 71 Euclid’s Postulate 4, 3, 88 Euclid’s Postulate 5, 3–4, 8–10, 322 implied by Euclidean parallel postulate, 190 implies Euclidean parallel postulate, 322 is independent, 134 Euclid’s segment cutoff theorem, 74 Euclidean area postulate, 200 Euclidean geometry, 53, 123, 185 is categorical, 132, 243 is consistent, 132 postulates of, 123, 382 Euclidean parallel postulate, 38, 185 implied by Euclid’s Postulate 5, 322 implies Euclid’s Postulate 5, 190 is independent, 133 Euclidean postulates, 321–333 Euler line theorem, 276 even integer, 434 even parity, 167 every triangle has two equal sides (fake theorem), 18 everywhere defined function, 428 exhaustion, method of, 279 existence and uniqueness, 398, 419 of parallels, 153 of two distinct points, 55 proof of, 417–419 statement of, 397 existence postulate, 55 existential quantifier, 397 existential statement, 397 extension of a field, 309 exterior of a circle, 248 of a polygon, 164, 167 of an angle, 95 exterior angle formed by a transversal, 150 of a convex polygon, 192 of a triangle, 113, 189 of an asymptotic triangle, 351 exterior angle inequality, 17, 113 for asymptotic triangles, 353 exterior angle sum for a convex polygon, 192 extreme and mean ratio, 225 extreme point, 65 weak, 174 fake theorem (every triangle has two equal sides), 18 family (synonym for set), 423 Fano plane, 28 Fermat prime, 318 Fermat, Pierre de, 318 field, 309 fifteen-sided polygon, 307 finite edge of an asymptotic triangle, 351 five-point plane, 27–29, 39 fixed point of a rigid motion, 443 foot of a perpendicular, 143 for all (universal quantifier), 394 four-point plane, 27–29, 38 four right angles theorem, 101 fourth angle of a Lambert quadrilateral, 342 fourth vertex of a Lambert quadrilateral, 342 free variable, 394 function, 24, 428 gaps in Euclid’s arguments, 13–18 Garfield, James A., 245 Gauss, Carl Friedrich, xiii, 10, 318 general polygonal region, 200 general triangle inequality, 116 geometric mean, 236 constructing, 301 Gergonne point, 276

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