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
Dividing a circle into areas where two of the old lines intersect. For each "O" and "I", the line "OI" crosses the circle in one point other than "O". Since the circle has infinitely many points, it has a point "A" which will be on none of the lines "OI". Then, for this point "A" and all of the old points "O", case "b" will be true. This lemma means that, if there are "k" lines crossing "AO", then each of them crosses "AO" at a different point and "k+1" new areas are created by the line "AO". The lemma establishes an important property for