Circles, a Mathematical View

ab,cd angle argument C₁ centre of inversion centre of similitude chapter circle centre circle of inversion circle orthogonal circle touches circle-tangential equation closed curve coaxal system determined complex numbers concentric circles concyclic conic system continuous function cross-ratio D[ab equal equilateral polygon Euclidean geometry excircles find the inverse fixed point given point h-point Hence inscribed intersecting type K₁ least upper bound limiting points line of centres locus M-transformation which maps midpoint Möbius transformation modulus nine-point circle non-Euclidean geometry obtain OX.OX Oxy-plane pairs parallel axiom passes perimeter perpendicular plane Oxy point of contact point of intersection point-circle polar plane problem of Apollonius proof prove quadric cone quartic curve radical axis radii radius ratio respect sa,bc solution straight line system of circles T₁ theorem three circles three given circles triangle ABC unique circle w-arc w-triangle x(tx