Book III. the circle, and DB touches it; the rectangle AD.DC is equal to the D square of DB. Let it pass 2 18. 3. E b 6. 2. d 1. 3. e 12. 1. Either DCA passes through the A But if DCA do not pass through the centre of the circle D F f 3. 3. IF from a point without a circle there be drawn two straight lines, one of which cuts the circle, and the other meets it; and if the rectangle contained by the whole line, which cuts the circle, and the part of it without the circle, be equal to the square of the line which meets it, the line which meets will touch the circle, Book III. a 17. 3. b 18. 3. c 36. 3. Let any point D be taken without the circle ABC, and from it let two straight lines DCA and DB be drawn, of which DCA cuts the circle, and DB meets it; if the rectangle AD.DC be equal to the square of DB, DB touches the circle. Drawa the straight line DE touching the circle ABC; find the centre F, and join FE, FB, FD; then FÉD is a rightb angle; and because DE touches the circle ABC, and DCA cuts it, the rectangle AD.DC is equal to the square of DE; but the rectangle AD.DC is, by hypothesis, B equal to the square of DB; therefore the square of DE is equal to the square of DB; therefore the F straight line DE is equal to the straight line DB. Because FE is equal to FB, and DE equal to DB, and the base FD common to the two triangles DEF, DBF, the angle DEF is equals to the angle DBF. But DEF is a right angle, therefore also DBF is a right angle. Now FB, if produced, will be a diameter, therefore DB touchese the circle ABC. Wherefore, if from a point, &c. Q. E. D. d 8. 1. e 16. 3. ELEMENTS OF GEOMETRY. BOOK IV. DEFINITIONS, I. other rectilineal figure, when all the angles of the inscribed II. scribed about another figure, when all III. ed in a circle, when all the angles of Book IV. IV. each side of the circumscribed figure v. scribed in a rectilineal figure, when VI. rectilineal figure, when the circumfe- VII. tremities of it are in the circumference of the circle. PROP. I. PROB. IN a given circle - to place a straight line equal to a given straight line, not greater than the diameter of the circle. |