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 on the point on the circle and rotate the compass. Apollonius of Perga showed that a circle may also be defined as the set of points in a plane having a constant "ratio" (other than 1) of distances to two fixed foci, "A" and "B". (The set of points where the distances are equal is the perpendicular bisector of segment "AB", a line.) That circle is sometimes said to be drawn "about" two points. The proof is in two parts. First, one must prove that, given two foci "A" and "B" and a ratio of distances, any point "P" satisfying