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 cross. In the second case (b), the new line crosses each of the old lines in a different point. It will be useful to know the following fact. Lemma. We can choose the new point "A" so that case "b" occurs for each of the new lines. Proof. Notice that, for the case "a", three points must be on one line: the new point "A", the old point "O" to which we draw the line, and the point "I" where two of the old lines intersect. Notice that there are "n" old points "O", and hence finitely many points "I"