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 a sphere Let "S" be a sphere with center "O", "P" a plane which intersects "S". Draw perpendicular to "P" and meeting "P" at "E". Let "A" and "B" be any two different points in the intersection. Then "AOE" and "BOE" are right triangles with a common side, "OE", and hypotenuses "AO" and "BO" equal. Therefore, the remaining sides "AE" and "BE" are equal. This proves that all points in the intersection are the same distance from the point "E" in the plane "P", in other words all points in the intersection lie on a circle "C" with center "E". This proves