Page:The Elements of Euclid for the Use of Schools and Colleges - 1872.djvu/136

112 PROPOSITION 37. THEOREM.

If from any point without a circle there he drawn two straight lines, one of which cuts the circle, and the other meets it, and if the rectangle contained in the whole line which cuts the circle, and the part of it without the circle, be equal to the square on the line which meets the circle, the line which meets the circle shall touch it.

Let any point D be taken without the circle ABC, and from it let two straight lines DCA, DB be drawn, of which DCA cuts the circle, and DB meets it; and let the rectangle AD, DC bw equal to the square on DB: DB shall touch the circle.

Draw the straight line DE, touching the circle ABC [III. 17. find F the centre, [III. 1. and join FB, FD, FE.

Then the angle FED is a right angle. [III. 18. And because DE touches the circle ABC, and DCA cuts it, the rectangle AD, DC is equal to the square on DE. [III. 36. But the rectangle AD, DC is equal to the square on DB. [Hyp. Therefore the square on DE is equal to the square on DB [Ax.1. therefore the straight line DE is equal to the straight line DB.

And EF is equal to BF; [I. Definition 15. therefore the two sides DE, EF are equal to the two sides DB, BF each to each; and the base DF is common to the two triangles DEF, DBF; therefore the angle DEF is equal to the angle DBF. [I. 8. But DEF is a right angle; [Construction. therefore also DBF is a right angle. And BF, if produced, is a diameter; and the straight line which is drawn at right angles to a diameter from the extremity of it touches the circle; [III. 16. Corollary. therefore DB touches the circle ABC.

Wherefore, if from a point &c.