96. If a semicircle be inscribed in a right-angled triangle so as to touch the hypothenuse and perpendicular, and from the end of its diameter a line be drawn through the point of contact to meet the perpendicular produced, the part produced will be equal to the perpendicular. 97. The circle described through any two of the angular points of a triangle and the intersection of the perpendiculars from the angles on the opposite sides will be equal to the circumscribing circle of the triangle. 98. AB, CD are chords of a circle, centre O, intersecting at right angles in E: shew that the squares of AB, CD together with four times the square of OE are double of the square of the diameter. 99. If ABCD be a parallelogram, and if a circle be described through A, cutting the sides AB, AD, and the diagonal AC, in F, H, G, respectively, then AB.AF+AD.AH AC.AĞ. 100. Given the vertical angle, the difference of the sides containing it, and the difference of the segments of the base made by a perpendicular from the vertex : construct the triangle. BOOK IV, DEFINITIONS. 1. 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. C 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. F 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, if it is not, BC is greater than D: make CE equal to D, and from the centre C, at the distance CE, describe the circle AEF, and join CA: Then, because C is the centre of the circle AEF, CA is equal to CE: But CE is equal to D; therefore CA is equal to D: Wherefore, in the circle ABC, a straight line is placed equal to the given straight line D, which is not greater than the diameter of the circle. Q. E. F. D B 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 DEF. B Draw the straight line GAH touching the circle in the point A (3. 17); and, at the point A, in the 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, and join BC: Then, because GAH touches the circle ABC, and AC is drawn from the point of contact cutting the circle, the angle HAC is equal to the angle ABC in the alternate segment of the circle (3. 32); but the angle HAC is equal to the angle DEF; therefore also the angle ABC is equal to the angle DEF: For the like reason, the angle ACB is equal to the angle DFE: Therefore the remaining angle BAC is equal to the remaining angle EDF: Wherefore the triangle ABC is equiangular to the given triangle DEF, and it is inscribed in the given circle ABC. Q. E. F. D E F 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 about the circle ABC a triangle equiangular to the triangle DEF. Produce EF both ways to the points G, H; find the centre K of the circle ABC, and from it draw any straight line KB; at the point K in the straight line BK, make the angle BKA equal to the angle DEG, and the angle BKC equal to the angle M B GE Η N D FH DFH; and through the points A, B, C, draw the straight lines LAM, MBN, NCL, touching the circle ABC (3,17). Then, because LM, MN, NL touch the circle ABC in the points A, B, C, to which from the centre are drawn KA, KB, KC, therefore the angles at the points A, B, C are right angles (3. 18): And because the four angles of the quadrilateral figure AKBM are equal to four right angles (for it can be divided into two triangles), and that two of them KAM, KBM are right angles, therefore the other two AKB, AMB are together equal to two right angles: But the angles DEG, DEF are also together equal to two right angles; therefore the angles AKB, AMB are equal to the angles DEG, DEF, of which AKB is equal to DEG, and therefore the remaining angle AMB, or LMN, is equal to the remaining angle DEF: And in like manner, the angle LNM may be shewn to be equal to DFE: And therefore the remaining angle MLN is equal to the remaining angle EDF: Wherefore the triangle LMN is equiangular to the given triangle DEF, and it is described about the given circle ABC. Q. E. F. PROP. IV. PROB. To inscribe a circle in a given triangle. Let ABC be the given triangle: it is required to inscribe a circle in the triangle ABC. Bisect the angles ABC, ACB by the straight lines BD, CD, meeting one another in the point D, from which draw DE, DF, DG perpendiculars to AB, BC, CA. Then, because the angle DBE is E equal to the angle DBF (for the angle ABC is bisected by BD), and that the B F G |