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; 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 shall touch the circle. 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 shall touch the circle. * * 1. Draw the straight line DE, touching the *17. 3. circle ABC, find its centre F, and join FE, †1.3. FB, FD: then FED is a right angle: and *18. 3 because DE touches the circle ABC, and DCA cuts it, the rectangle AD, DC is equal to the *35. 3. 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 † 1 Ax. of DB; and the straight line DE equal to the straight line DB: and FE is equal to FB; +15Def. wherefore DE, EF are equal to DB, BF, each to each and the base FD is common to the two triangles DEF, DBF; therefore the angle DEF is equal * to the angle DBF: but DEF was shown to B be a right angle; 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 D *8. 1. E †1 Ax. the extremity of it, touches the circle: there- * 16. 3. fore DB touches the circle ABC. Wherefore, if from a point, &c. Q. E.D. 106 THE ELEMENTS OF EUCLID. BOOK IV. DEFINITIONS. I. A RECTILINEAL figure is said to be inscribed in another 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 described 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. IV. A rectilineal figure is said to be described about a circle, when each side of the circumscribed figure touches the circumference of the circle. V. In like manner, a circle is said to be inscribed in a rectilineal figure, when the circumference of the circle touches each side of the figure. VI. A circle is said to be described about a rectilineal figure, when the circumference of the circle passes through all the angular points of the figure about which it is described. 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 which is 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: it is required to place in the circle ABC a straight line equal to D. *3. 1. Draw BC the diameter of the circle ABC; then, if BC is equal to D, the thing required is done; for in the circle ABC a straight line BC is placed equal to D: but, * +Hyp. if it is not, BC is greater+ 1. + Const. join CA: CA shall be equal to D. B Because C is the centre of the circle AEF, +15 Def. CA is equal to CE: but D is equal to CE; therefore D is equal to CA. Wherefore in the circle ABC, a straight line CA is placed equal to the given straight line D, which is not greater than the diameter of the circle. Which was to be done. +1 Ax. * 17.3. PROP. II. PROB. In a given circle to inscribe a triangle equiangular to a given triangle. Let ABC be the given circle, and DEF the given triangle; it is required to inscribe in the circle ABC a triangle equiangular to the triangle DEF. G D Draw the straight line GAH touching the circle in the point A, and at the point A, in the *23. 1. straight line AH. make* the angle HAC equal to the angle DEF; and at the point A, in the straight line AG, make the angle GAB equal to the angle DFE; E and join BC: ABC +1.3. shall be the triangle required. B H Find the centre, and through it draw any straight line +17 Def. BC terminated both ways by the circumference; this line + is 1. a diameter, Because HAG touches the circle ABC, and AC is drawn from the point of contact, the angle * HAC is equal to the angle ABC in the alter- *32. 3. nate segment of the circle: but HAC is equal + + Const. to the angle DEF; therefore also the angle ABC is equal to DEF: for the same reason, the †1 Ax. angle ACB is equal to the angle DFE: therefore the remaining angle BAC is equal* to the *32. 1. remaining angle EDF: wherefore the triangle ABC is equiangular to the triangle DEF, and it is inscribed in the circle ABC. Which was to be done. PROP. III. PROB. About a given circle to describe a triangle equiangular to a given triangle. Let ABC be the given circle, and DEF the given triangle; it is required to describe a triangle about the circle ABC equiangular to the triangle DEF. and 1 Ax. Produce EF both ways to the points G, H; find the centre K of the circle ABC, and from +1.5 it draw any straight line KB; at the point K in the straight line KB, make* the angle BKA *23. 1. equal to the angle DEG, and the angle BKC equal to the angle DFH; and through the points A, B, C, draw the straight lines LAM, MBN, NCL, touching the circle ABC: LMN *17. 3. shall be the triangle required. * Because LM, MN, NL touch the circle ABC in the points A, B, C, to which from the centre are drawn KA, KB, KC, the angles at the points A, B, C are right* angles: and because *18. 3. the four angles of the quadrilateral figure AMBK are equal to four right angles, ‡ for it can be + Suppose the line MK drawn: then, because the three angles of every triangle are together equal to two right angles; †32. L therefore all the angles of the two triangles AMK, BMK are together equal to four right angles; but all the angles of these triangles are together equal to the angles of the figure AMBK therefore all the angles of the figure AMBK are together equal to four right angles. L |