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1. Draw a diameter BC.
2. From B, with radius BA, cut the circumference in D
and E. 3. From D, E, and C as centres, with DE as radius,
describe arcs intercepting in G, F, and H. 4. Join GF, FH, and HG; then FGH is the required equi
lateral triangle described about the given circle A.
To describe a triangle about a given circle O, having angles equal to those of a given triangle A BC.
1. Produce any side of the triangle, as BC, both ways to D
2. Draw any radius OF, and draw GH as a tangent through
F (Pr. 54).
To inscribe a square in a given regular pentagon ABCDE.
1. Draw BE and BF at right angies to BE, and equal to it
3. Draw GH parallel to BF, and HK parallel to BE
(Pr. 8). 4. Draw KL parallel to HG, and GL parallel to HK; then
HGLK will be the required square, inscribed in the given pentagon ABCDE.
Problem 102. To inscribe a square within a given regular hexagon ABCDEF.
1. Draw a diagonal FC, and bisect FC by a perpendicular
(Pr. 1) in G, and let the line of bisection cut the
hexagon in H and K. 2. Bisect any two adjacent angles, as FGH, CGH (Pr. 4), and produce the lines of bisection to meet the hexagon in L, M, N, 0.
3. Join LN, NM, MO, and OL by straight lines, and the
figure LMNO is the required square, inscribed in the given hexagon ABCDEF.
Problem 103. To inscribe a square within a given quadrant ABC, two of its corners being in the arc.
1. Draw the chord BC, and at one of the extremities, say B,
draw BD perpendicular and equal to it (Pr. 2).
2. Draw the line DA, cutting the arc BC in E.
EF is a side of the required square. Complete the square
within the given quadrant ABC. Note. The same method is to be observed in inscribing a square in any sector of a circle (acute-angled or obtuse-angled).
Problem 104. To inscribe a four-sided equilateral figure in any given parallelogram ABCD.
1. Draw the diagonals AD, BC, cutting each other in E.
2. Bisect any two of the adjacent angles at E (Pr. 4),
by lines cutting the sides of the parallelogram in
F, G, H, K. 3. Join HF, FK, &c., and GHFK will be a four-sided equi
lateral figure, inscribed in a given parallelogram ABCD.
Problem 105. To inscribe a rectangle in a given triangle ABC, haring a side equal to a given line DĖ.
1. On BC, mark off BF equal to DE.
draw GH parallel to BC (Pr. 9).
3. From G and H, draw GK and HL perpendicular to the
base BC (Pr. 3); then HGKL is the required rectangle, and it is inscribed in the given triangle ABC.
1. Draw the diagonals AD, BC, intersecting each other
2. From A, B, C, and D, with CE as radius, describe quad
rants cutting the sides of the square in F, G, H, K, L,
M, N, 0. 3. Join these points, and the required octagon will be
inscribed in the given square ABCD.