EXAMPLE. Make a triangle, A B C, having the sides AB, AC, BC, respectively equal to 5 inches, 3 inches, and 4 inches, and from the point A, describe arcs touching the side B C in the points B and C. PROBLEM XXVIII. To inscribe a circle in any triangle. 1. Bisect any two angles, as CAB, CBA, by the lines AD, BD, cutting each other in the point D. A E B 2. From the point D, draw D E perpendicular to the line AB. From D as a centre, with DE as a radius, describe a circle, which will be inscribed in the triangle, i. e. all the sides of the triangle will be tangents to the circle. EXAMPLES. 1. Inscribe a circle in any obtuse-angled triangle, and examine the tangents. 2. Describe an equilateral triangle, having each of its sides 2 inches. Inscribe a circle in it, and from its centre, draw perpendiculars to each of its sides. If correct, the sides of the triangle will be bisected by the perpendiculars let fall on them. PROBLEM XXIX. To inscribe a square in a given circle. 1. In the given circle draw two diameters at right angles to each other, which will divide the circumference into four equal arcs. 2. Join the extremities of each arc by a chord, when the four chords will form the square required, which will be inscribed in the given circle, as each of its angular points are in the circumference. PROBLEM XXX. To describe, or circumscribe, a circle about a given square. 1. In the given square, draw its two diagonals, or diameters. 2. From the point of intersection of these diagonals, as a centre, with the distance from this point to either of the angular points of the given square, as a radius, describe a circle, which will pass through all the other angular points of the square, and thus be described about it. PROBLEM XXXI. To describe a square about a given circle. 1. In the given circle draw two diameters, at right angles to each other. 2. At the extremities of these diameters draw tangents to the circle, (Prob. XXII.,) which, being produced till they meet, will form the square required. EXAMPLE. Describe a square about a circle whose radius is one inch. Bisect each of the four circumferences, formed in the construction. Join these points of bisection, when a square will be inscribed in the circle, having its sides respectively parallel to the circumscribed square. PROBLEM XXXII. To inscribe a circle in a square. 1. Bisect each of the sides of the given square. Join the points of bisection by lines cutting each other. 2. From this point of intersection, as a centre, with a radius extending to either extremity of one of the lines of bisection, describe a circle, which will pass through the other points, touching the sides of the given square, and thus be inscribed in it. E EXAMPLE. Make a square having each side 14 inches, and describe one circle about, and inscribe another in it. PROBLEM XXXIII. To trisect a right angle, that is, to divide it into three equal parts. Let A B C be the given right angle. 1. From B as a centre, with any radius describe an arc, cutting the lines BA and BC; and from the points of intersection as centres, with the same radius as before, cut the arc in D and E. 2. Draw the lines D B, E B, when the angle A B C will be trisected, as required. A Q B This problem illustrates the valuable geometrical fact, that the radius will cut off from the circumference of any circle, an arc of 60 degrees, or one-sixth of its circumference; the problem is, therefore, applicable in many cases where the division of an arc or angle is required. For example, let it be required to cut off 15 degrees from any arc: find (Prob. XXIV.) the centre, set off 60 degrees, by application of the radius; and, by two successive operations of Prob. VIII., divide that portion into four equal parts; each part will contain 15 degrees. No method purely geometrical has hitherto been discovered, by which any other than a right angle can be trisected. EXAMPLES. 1. Find the twelfth part of the circumference of a circle of 11⁄2 inches radius. 2. Make, by this problem, an angle of 75°, and verify the result by the scale of chords. 3. Make an angle of 17230, and verify by the protractor. PROBLEM XXXIV. On any straight line, as a base, to make an isosceles triangle, that shall have a given vertical angle. Let A B be the given base. 1. Produce A B either way, as towards D; and from A, draw A C, making the angle CAD equal to the given angle. 2. Bisect the angle C A B by the D line AE: from B, draw B E, making the angle ABE equal to the angle BAE, when the angle A E B will be equal to the angle CA D, and AEB will be the triangle required. EXAMPLES. 1. On a line 2 inches long, describe an isosceles triangle, with its vertical angle containing 127 degrees. 2. On a base of 14 inches, describe an isosceles triangle, having its vertical angle 45 degrees. Bisect this angle by a line meeting the base, and show that this line bisects, and is perpendicular to, the base of the triangle. PROBLEM XXXV. An acute-angled isosceles triangle being given, to describe another on its base, whose vertical angle shall be double of that of the given triangle. Let ACB be the given triangle. 1. Bisect the base AB in H, and draw C H. 2. Draw A D, making the angle D A C equal to the angle A CD, and meeting CH in the point D. 3. Draw DB, and ADB will be the triangle required. Α. H B EXAMPLE. Make (Prob. XXXIV.) an isosceles triangle, having its base 1 inches, and its vertical angle 72°, and on the same base, another, with its vertical angle 144°. PROBLEM XXXVI. An isosceles triangle being given, to describe another on the same base, whose vertical angle shall be half that of the given triangle. Let ADB (see last Fig.) be the isosceles triangle, of which A D B is the vertical angle given. Bisect A B in H. Draw HD and produce it towards C. Make DC equal to DA or DB. Draw CA and CB, and ACB will be the isosceles triangle required. EXAMPLES. 1. Given an isosceles triangle, whose base is 1 inch, and its other sides inch each; to make another triangle on the same base with its vertical angle half of that of the given triangle. 2. Make an isosceles triangle, whose base is 1 inches, and vertical angle 120°. On the opposite side of this base, make another triangle whose vertical angle is 60°. Describe a circle about this last triangle, and its circumference shall pass through the vertex of the first triangle. PROBLEM XXXVII. On a straight line, to describe a segment of a circle, that shall contain an angle equal to a given angle. Let A B be the given straight line. 1. Make BAC equal to the given angle. 2. Make the angle A B C equal to the angle BAC. Draw AD and BD at right angles to AC and B C respectively, meeting in the point D. B |