Book 3
16
Porism. From this it is manifest that the straight line drawn at right angles to the diameter of a circle from its extremity touches the circle. Q. E. D.
PROPOSITION 17.
17
From a given point to draw a straight line touching a given circle.
17
Let A be the given point, and BCD the given circle; thus it is required to draw from the point A a straight line touching the circle BCD.
17
For let the centre E of the circle be taken; [III. 1] let AE be joined, and with centre E and distance EA let the circle AFG be described; from D let DF be drawn at right angles to EA, and let EF, AB be joined; I say that AB has been drawn from the point A touching the circle BCD.
17
For, since E is the centre of the circles BCD, AFG, EA is equal to EF, and ED to EB; therefore the two sides AE, EB are equal to the two sides FE, ED: and they contain a common angle, the angle at E; therefore the base DF is equal to the base AB, and the triangle DEF is equal to the triangle BEA, and the remaining angles to the remaining angles; [I. 4] therefore the angle EDF is equal to the angle EBA.
17
But the angle EDF is right; therefore the angle EBA is also right.
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.
