Problem 93. To inscribe an equilateral triangle in a given square ABCD. B H C D 1. From C, with radius AC, describe the quadrant AD. 2. From A and D, with the same radius, cut off AE and DF. 3. Bisect AF and ED, and through the points of bisection draw the lines CG, CH, cutting the sides of the square in G, H. 4. Draw GH; then GCH is the required equilateral triangle, inscribed in the given square ABCD. Problem 94. To inscribe an equilateral triangle in a given hexagon, so that its sides are parallel to three sides of the hexagon. B 1. Bisect the alternate sides of the given hexagon (Pr. 1) in the points A, B, and C. 2. Join these points, and an equilateral triangle will be inscribed in the given hexagon. NOTE.—By joining the three alternate angles of the hexagon, the largest equilateral triangle it will contain will be inscribed. Problem 95. To inscribe an equilateral triangle in a given regular pentagon ABCDE. 1. From A as centre, with any radius, describe a semicircle FGH. 2. From F and with the same radius, describe arcs cut ting the semicircle in K and L. 3. From A, draw lines through K and L, meeting the sides of the pentagon in M and N respectively. 4. Join MN, and AMN will be the required equilateral tri angle, inscribed in the given pentagon ABCDE. Problem 96. To inscribe an isosceles triangle within a given square ABCD, having a given base EF. 1. Draw a diagonal BC, and bisect EF in G (Pr. 1). or GF 3. With H as centre, and HB radius, cut the sides of the square AB and BD in the points K and L. 4. Join CK, KL, and LC, and an isosceles triangle CKL will be inscribed within the given square ABCD. Problem 97. To inscribe a square within a given circle A. B E 1. Find the centre of the circle A (Pr. 45). DE. inscribed within the given circle A. Problem 98. To inscribe a square within a given triangle ABC. 1. Draw AD, the altitude of the given triangle (Pr. 21). it equal to the base BC. 3. Draw the line ED, cutting AC in F. (Pr. 3); then FG is one side of the required square. and from H, with the same length, cut AB in K. 6. Join HK, KF; then KFGH is a square inscribed in the given triangle A BC. Problem 99. To inscribe a square within a given rhombus ABCD. 1. Draw the two diagonals AC, BD. EF, GH, cutting the sides of the rhombus in K and L. 3. Join FH, HK, KL, and LF; then FHKL is the required square, inscribed within the given rhombus ABCD. To inscribe a square in a given trapezium ABCD, which has its adjacent pairs of sides equal. 1. Draw a diagonal BD, bisecting the trapezium and the angle at B. K H 2. Find the centre of the figure in point E by bisecting another angle, as at C (Pr. 4). 3. At point E raise a perpendicular to BD, as EF. 4. Bisect the right angles on either side of EF, and produce the lines of bisection to cut the trapezium in GĖKL. 5. Join GK, KH, HL, and LG, and the figure GKHL will be the required square, inscribed in the given trapezium ABCD. |