Proof by contradiction

A theorem can be proved by contradiction. First assume that the conclusion is wrong ("Suppose it is not so"). From that assumption, by logic, reach a fact that contradicts an axiom, a theorem or the condition ("We have a contradiction").

The conclusion: our assumption is wrong, so the theorem is true. Theorem 13.1 is proved like this: two lines perpendicular to a third line are parallel. Suppose a is perpendicular to c and b is perpendicular to c, but a and b meet at M. Then two lines through M are perpendicular to c. This contradicts the fact that only one line through a point is perpendicular to a given line.

The same way: theorem 13.2 (a contradiction with the parallel axiom: through a point not on a line there is only one line parallel to it) and theorem 21.3 (the radius to the point of tangency is perpendicular to the tangent: a perpendicular OM would be longer than a slant OA).