Book III. the circle, and DB touches it; the rectangle AD.DC is equal to the square of DB. D a 18.3. b 6. 2. с 47. 1. Either DCA passes through the C E But if DCA do not pass through the centre of the circle d 1. 3. ABC, taked the centre E, and draw D AD.DC+EC2=ED2. Now ED2=EB2+BD2=EC2+BD2 ; therefore AD.DC+EC2=EC2+BD2; therefore AD.DC= BD2. Wherefore, if from any point, &c. Q. E. D. COR. If from any point without a circle there be drawn two straight lines cutting it, as AB, AC, the rectangles contained by the whole lines and the parts of them without the circle are equal to each other, BA.AE=CA.AF; for each of these D rectangles is equal to the square of the straight line AD which touches the circle. E C B PROP. XXXVII. THEOR. 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. Book III. 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. a 17.3. b 18. 3. c 36. 3. Drawa the straight line DE touching the circle ABC; find the centre F, and join FE, FB, FD; then FED is a right angle; and because DE touches the circle ABC, and DCA cuts it, the rectangle AD.DC is equale to the square of DE; but the rectangle AD.DC is, by hypothesis, equal to the square of DB; therefore the square of DE is equal to the square of DB; therefore the 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 D C B E F A d 8. 1. e 16. 3. triangles DEF, DBF, the angle DEF is equald 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 touches the circle ABC. Wherefore, if from a point, &c. Q. E. D. ELEMENTS OF GEOMETRY. BOOK IV. DEFINITIONS. I. A RECTILINEAL figure is said to be inscribed in an- Book IV. other rectilineal figure, when all the angles of the inscribed figure are upon the sides of the figure in which it is inscribed, each upon each. II. In like manner, a figure is said to be de scribed about another figure, when all the sides of the circumscribed figure pass through the angular points of the figure about which it is described, each through each. III. A rectilineal figure is said to be inscribed in a circle, when all the angles of the inscribed figure are upon the circumference of the circle. A rectilineal figure is said to be described about a circle, when each side of the circumscribed figure V. In like manner, a circle is said to be in- VI. A circle is said to be described about a VII. A straight line is said to be placed in a circle, when the extremities 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. Let ABC be the given circle, and D the given straight line, not greater than the diameter of the circle. Draw BC the diameter of the circle ABC; then, if BC be equal to D, the thing required is done; for in the circle ABC a straight line BC is placed equal to D. But if D A C B |