Page:The Elements of Euclid for the Use of Schools and Colleges - 1872.djvu/114

90 Wherefore, the straight line &c.

From this it is manifest, that the straight line which is drawn at right angles to the diameter of a circle from the extremity of it, touches the circle; [III. Def. 2. and that it touches the circle at one point only,

because if it did meet the circle at two points it would fall within it. [III. 2.

Also it is evident, that there can be but one straight line which touches the circle at the same point.

PROPOSITION 17. PROBLEM.

To draw a straight line from a given point, either without or in the circumference, which shall touch a given circle.

First, let the given point A be without the given circle BCD: it is required to draw from A a straight line, which shall touch the given circle.

Take E, the centre of the circle, [III. 1.

and join AE cutting the circumference of the given circle at D;

and from the centre E, at the distance EA, describe the circle AFG; from the point D draw DF at right angles to EA,[I.11.

and join EF cutting the circumference of the given circle at B;

join AB. AB shall touch the circle BCD.

For, because E is the centre of the circle AFG, EA is equal to EF. [I. Definition 15. And because E is the centre of the circle BCD, EB is equal to ED. [I. Definition 15.

Therefore the two sides AE, EB are equal to the two sides FE, ED, each to each;

and the angle at E is common to the two triangles AEB, FED;

therefore the triangle AEB is equal to the triangle FED, and the other angles to the other angles, each to each, to which the equal sides are opposite; [I. 4.