If a straight line touch a circle, and a straight line be joined from the centre to the point of contact, the straight line so joined will be perpendicular to the tangent.
For let a straight line DE touch the circle ABC at the point C, let the centre
F of the circle ABC be taken, and let FC be joined from F to C; I say that FC
is perpendicular to DE.
For, if not, let FG be drawn from F perpendicular to DE.
Then, since the angle FGC is right, the angle FCG is acute;
and the greater angle is subtended by the greater side; therefore FC is
greater than FG.
But FC is equal to FB; therefore FB is also greater than FG, the less than the
greater: which is impossible.
Therefore FG is not perpendicular to DE.
Similarly we can prove that neither is any other straight line except FC;
therefore FC is perpendicular to DE.
Therefore, etc.
Q.E.D.