A1∠BAC = ½ arc BC
an inscribed angle is half the central angle — arc BC is the arc not containing A (Proof 51)
A2the two arcs sum to 360°
the two arcs cut off by two points on a circle (Proof 51)
A3same side ⟺ same arc
the line through chord BC divides the remaining points of the circle into two arcs, and the points on the same side of that line form one arc
A4a segment meeting ℓ ⟹ opposite sides
plane separation — if it does not meet, the same side
A5two points determine one line
determination of a line — two distinct lines meet in at most one point
A6vertical angles are equal
Proof 48
A7opposite rays ⟹ sum 180°
straight angle — when A is between P and B, ∠PAC + ∠CAB = 180°
A8two pairs of equal angles ⟹ similar
AA similarity condition (Proof 53)
A9similar ⟹ corresponding sides proportional
definition of similarity (Proof 53)
A10a/b = c/d ⟺ ad = bc
property of proportions (b, d ≠ 0)
A11length of a segment > 0
positivity of length
A12associativity, commutativity, distributivity
arithmetic of the reals
A13tangent-chord angle = inscribed angle
the angle between a tangent and a chord equals the inscribed angle on that chord (Euclid III.32)
A14P outside ⟹ A between P and B
convexity of the disk — for a secant through an exterior point, name the nearer intersection A
A15P inside ⟹ P between A and B
an interior point lies between the two intersections
A16betweenness ⟹ same ray
if Q is between P and R, then ray PQ and ray PR are the same
Rassumption contradictory ⟹ negation true
proof by contradiction (Proof 26)
Ea = b ⟹ same operation on both sides
properties of equality
Nothing outside this list may be cited.