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

Let *A* be the given point, and *BCD* the given circle.

It is required to draw from the point *A* a straight line touching the circle *BCD.*

Take the center *E* of the circle, and join *AE.* Describe the circle *AFG* with center *E* and radius *EA.* Draw *DF* from *D* at right angles to *EA.* Join *EF* and *AB.*

I say that *AB* has been drawn from the point *A* touching the circle *BCD.*

For, since *E* is the center of the circles *BCD* and *AFG, EA* equals *EF,* and *ED* equals *EB.* Therefore the two sides *AE* and *EB* equal the two sides *FE* and *ED,* and they contain a common angle, the angle at *E,* therefore the base *DF* equals the base *AB,* and the triangle *DEF* equals the triangle *BEA,* and the remaining angles to the remaining angles, therefore the angle *EDF* equals the angle *EBA.*

But the angle *EDF* is right, therefore the angle *EBA* is also right.

Now *EB* is a radius, and the straight line drawn at right angles to the diameter of a circle, from its end, touches the circle, therefore *AB* touches the circle *BCD.*

Therefore from the given point *A* the straight line *AB* has been drawn touching the circle *BCD.*

Q.E.F.