Kitap 3
17
Now EB is a radius; and the straight line drawn at right angles to the diameter of a circle, from its extremity, touches the circle; [III. 16, Por.] therefore AB touches the circle BCD.
17
Therefore from the given point A the straight line AB has been drawn touching the circle BCD.
PROPOSITION 18.
18
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.
18
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.
18
For, if not, let FG be drawn from F perpendicular to DE.
18
Then, since the angle FGC is right, the angle FCG is acute; [I. 17] and the greater angle is subtended by the greater side; [I. 19] therefore FC is greater than FG.
18
But FC is equal to FB; therefore FB is also greater than FG, the less than the greater: which is impossible.
18
Therefore FG is not perpendicular to DE.
18
Similarly we can prove that neither is any other straight line except FC; therefore FC is perpendicular to DE.
18
Therefore etc. Q. E. D. the tangent, ἡ ἐφαπτομένη.
PROPOSITION 19.
19
If a straight line touch a circle, and from the point of contact a straight line be drawn at right angles to the tangent, the centre of the circle will be on the straight line so drawn.
