458 Index concave polygon, 158 concave vertex, 172 concentric circles, 247, 248 conclusion in Greek proofs, 7 of a rule of inference, 408 of an implication, 388 concurrence theorem altitude, 269 angle bisector, 268 median, 224 perpendicular bisector, 264 concurrent, 223, 262 conditional statement, 388 congruence in Hilbert’s axioms, 20, 23, 375, 376 of angles, 88 of asymptotic triangles, 351 of convex polygons, 159 of polygons, 173 of segments, 64 of simple polygonal regions, 199 of triangles, 106 via rigid motions, 446 congruent, see congruence conjugate arcs, 254, 257 conjunction, 385 connective, logical, 385 consecutive angles of a polygon, 156 consecutive interior angles, 150 consecutive interior angles theorem, 151 converse, 186 consecutive vertices of a polygon, 156 consistency of an axiomatic system, 32 of betweenness of points, 61 of betweenness of rays, 90 of endpoints of arcs, 256 of endpoints of rays, 76 of endpoints of segments, 66 of polygon vertices, 157, 159 of triangle vertices, 103 of vertices of proper angles, 84 relative, 32 constant of proportionality, 214 constructible angle, 313 constructible circle, 308 constructible line, 308 constructible number, 308 characterization of, 311 constructible point, 308 constructing a perpendicular, 101 constructing a rectangle, 195 constructing a square, 196 construction compass and straightedge, 296–319 in Greek proofs, 3, 6, 7 constructive proof, 417 contains (a point) in incidence geometry, 24 in plane geometry, 55 contrapositive, 391 proof by, 411 converse, 391 convex decomposition lemma, 201 convex polygon, 157–166 angle criterion, 160 characterizations of, 173 diagonal criterion, 176 interior of, 164, 171 semiparallel criterion, 161, 176 vertex criterion, 160 convex quadrilateral, 175–183 diagonal criterion, 176 semiparallel criterion, 176 convex set, 80 intersection of, 80 convex vertex, 172 coordinate of a point, 57 of a ray, 86 coordinate function adapted to a ray, 72 for a half-rotation, 86 for a line, 57 starting at a ray, 86 coordinate representation of a ray, 72 of a segment, 67 copying theorem asymptotic triangle, 352 quadrilateral, 179, 300 triangle, 111, 300 corollary, 6, 24 correspondence, one-to-one, 429 corresponding angles, 150 corresponding angles theorem, 151 converse, 186 cosine, 238, 239 is injective, 239 cosines, law of, 240 cross lemma, 177

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