Proposition 17

From a given point to draw a straight line touching a given circle.

Let TeX Embedding failed! be the given point and TeX Embedding failed! the given circle;
 thus it is required to draw from the point TeX Embedding failed! a straight line touching the circle TeX Embedding failed!.

For let the centre TeX Embedding failed! of the circle be taken; [Prop 3.1]

let TeX Embedding failed! be joined, and with centre TeX Embedding failed! and distance TeX Embedding failed! let the circle TeX Embedding failed! be described;

from TeX Embedding failed! let TeX Embedding failed! be drawn at right angles to TeX Embedding failed!,

and let TeX Embedding failed!, TeX Embedding failed! be joined;

I say that TeX Embedding failed! has been drawn from the point TeX Embedding failed! trouching the circle TeX Embedding failed!.

For, since TeX Embedding failed! is the centre of the circles TeX Embedding failed!, TeX Embedding failed!,

TeX Embedding failed! is equal to TeX Embedding failed!, and TeX Embedding failed! to TeX Embedding failed!;

therefore the two sides TeX Embedding failed!, TeX Embedding failed! are equal to the two sides TeX Embedding failed!, TeX Embedding failed!:

and they contain a common angle, the angle at TeX Embedding failed!;

therefore the base TeX Embedding failed! is equal to the base TeX Embedding failed!,

and the triangle TeX Embedding failed! is equal to the triangle TeX Embedding failed!,

and the remaining angles to the remaining angles; [Prop 1.4]

therefore the angle TeX Embedding failed! is equal to the angle TeX Embedding failed!.

But the angle TeX Embedding failed! is right;

therefore the angle TeX Embedding failed! is also right.

Now TeX Embedding failed! is a radius;

and the straight line drawn at right angles to the diameter of a circle, from its extremity, touches the circle; [Prop 3.16. Porism]

therefore TeX Embedding failed! touches the circle TeX Embedding failed!.

Therefore from teh given point TeX Embedding failed! the straight line TeX Embedding failed! has been drawn touching the circle TeX Embedding failed!.