*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!*.