Secant line point a "tangent line" and in no points an "exterior line". A "chord" of a circle is the line segment that joins two distinct points of the circle. A chord is therefore contained in a unique secant line and each secant line determines a unique chord. In rigorous modern treatments of plane geometry, results that seem obvious and were assumed (without statement) by Euclid in his treatment, are usually proved. For instance, "Theorem (Elementary Circular Continuity)": If formula_1 is a circle and formula_2 a line that contains a point that is inside formula_1 and a point that is outside of
Circle of antisimilitude also orthogonal to both circles of antisimilitude. Suppose that, for three circles "α", "β", and "γ", there is a circle of antisimilitude for the pair ("α","β") that crosses a second circle of antisimilitude for the pair ("β","γ"). Then there is a third circle of antisimiltude for the third pair ("α","γ") such that the three circles of antisimilitude cross each other in two triple intersection points. Altogether, at most eight triple crossing points may be generated in this way, for there are two ways of choosing each of the first two circles and two points where the two chosen circles cross.