A circle is a plane figure contained by a single line [which is called a circumference], (such that) all of the straight lines radiating towards [the circumference] from one point amongst those lying inside the figure are congruent to one another.
Let \(A,B\) be two different points in a plane \(\mathcal P\), and let \(d(A,B)\) denote their Euclidean distance (i.e. the length of the segment \(\overline{AB}\)). A circle is a plane figure consisting of points of the plane \(D\), which have a smaller or equal distance from \(A\), formally
\[\text{Circle}:=\{D\in\mathcal P:~d(A,D)\le d(A,B)\}.\]
Any segment \(\overline{A,D}\) with maximum possible distance \(d(A,D)=d(A,B)\) is called the radius of the circle.
The circumference of the circle is the boundary of the circle, i.e. all points \(D\) in the plane, which have the maximum possible distance:
\[\text{Circlumference}:=\{D\in\mathcal P:~d(A,D) = d(A,B)\}.\]
A circle with a radius \(\overline{AB}\). In the figure, one of the infinitely many points \(D\) with \(d(A,D)\le d(A,B)\) is marked.
Axioms: 1
Chapters: 2
Corollaries: 3 4 5 6 7
Definitions: 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28
Lemmas: 29
Problems: 30 31
Proofs: 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116
Propositions: 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177
Solutions: 178 179
Topics: 180 181