Theorem 3.7: A tangent line to a circle is perpendicular to the radius to the point of tangency.
Proof: Suppose that the line from the center of the circle to the point of intersection is not perpendicular to the alleged tangent line.
Let A be the point where the line intersects the circle, and let O be the center of the circle. If the line from O to A is not perpendicular to the line, then let F be the foot of the center in the line. Let B be the point which is on the line on the other side of F from A whose distance from F is the same as the distance from F to A. Such a point can be obtained by letting
where
By Theorem 2.2, |B, F| = |A, F|, and B is on the other side of F from A by definition.
Since AB is perpendicular to OF at F, we can use the Pythagorean theorem (Theorem 3.4) to conclude that
B is another point where the line intersects the circle, which contradicts the fact that the line is tangent to the circle at A.