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 satisfy The intersection of the spheres is the set of points satisfying both equations. Subtracting the equations gives In the singular case formula_7, the spheres are concentric. There are two possibilities: if formula_8, the spheres coincide, and the intersection is the entire sphere; if formula_9, the spheres are disjoint and the intersection is empty. When "a" is nonzero, the intersection lies in a vertical plane with this x-coordinate, which may intersect both of the spheres, be tangent to both spheres, or external to both spheres. The result follows from the previous proof for sphere-plane intersections. Circle of a sphere A