Exercise 4.3.6
Conjecture: If a line is tangent to a circle, then the line is perpendicular to a radius of the circle at the point of tangency.
Contrapositive of Conjecture: If a line that intersects a circle is not perpendicular to a radius of the circle at the point of intersection, then the line is not tangent (that is- the line intersects at more than one point of the circle).
Proof (By contrapositive): Construct the circle centered at point O. Now, consider the line l that intersects circle O at the point P. By Euclid's Proposition 13, we know that the radius at point P forms two angles, call them and such that .
Assume that . That is, assume l is not perpendicular to a radius of the circle at P. Then, without loss of generality, we can also assume that .
Now, form the isosceles triangle such that Q lies on l, the angle coincides with angle , and . By Euclid's Proposition 6, the segment must equal the length of the radius. Thus, the point Q must be a point on the circle. Then, we know that l intersects the circle at P and Q. Because l intersects the circle at more than one point, l is not tangent to the circle.
Therefore, the contrapositive of the conjecture above is true and thus the conjecture is also true.