A I. Right lined figure is faid to be infcribed in a right lined Book IV, figure, when every one of the angles of the inscribed fi gure touches every one of the fides of the figure wherein it is infcribed. II. A right lined figure is faid to be described about a right lined figure, when every one of the fides of the circumfcribed figure touches each of the angles of the right lined figure. III. A right lined figure is infcribed in a circle when each of the angles of the infcribed figure touches the circumference of the circle. IV. A right lined figure is defcribed about a circle when each of the fides of the circumfcribed figure touches the circumference of the circle. V. A circle is infcribed in a right lined figure, when the circumference of the circle touches all the fides of the figure in which it is infcribed. VI. A circle is defcribed about a right lined figure when the circumference of the circle touches all the angles of the figure. VII. A A right line is applied in a circle when its extremes are in the circumference of the circle. 23. 1. PROP. I. PRO B. To apply a right line in a circle, equal to a given right line, not greater than the diameter of the circle. It is required to apply a right line in the circle ABC, equal to a given right line D, not greater than the diameter of the circle. Draw the diameter BC; if equal to D, what was required is done; if not, the diameter BC is greater than D; put CE equal to Da ; about the center C, with the diftance CE, describe the circle AEF; then CA is equal to CE; but CE is equal to D; therefore CA is equal to D. Wherefore, there is drawn, &c. 2 17. 3. B23, Ia £ 32. 3. PROP. II. PROB. IN It is required to inscribe a triangle, in a given circle ABC, equiangular to a given triangle DEF: Draw the right line GAH, touching the circle in the point A, with the right line AH, at the point A, make the angle HAC equal to the angle DEF, and the angle GAB equal to DFE; join BC. Then the angle HAC is equal to the angle ABC; but the angle HAC is equal to DEF; therefore, ABC is equal to DEF : And BAG is equal to ACB; but BAG is equal to DFE; therefore ACB, is equal to DFE; therefore the remaining third d Cor. 32. angles BAC, EDF are equal; therefore the triangle ABC, is inscribed in the circle ABC, and equiangular to the triangle DEF, which was required. Wherefore, &c. d PROP. |