⁦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).