Problem 130.

To describe an equilateral triangle about a given circle A. 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 equilateral triangle described about the given circle A.

Problem 131.

To describe a triangle about a given circle O, having angles equal to those of a given triangle A BČ.

1. Produce any side of the triangle, as BC, both ways to D and E.

2. Draw any radius OF, and draw GH as a tangent through

F (Pr. 54).

Problem 101.

To inscribe a square in a given regular pentagon ABCDE.

1. Draw BE and BF at right angles to BE, and equal to it (Pr. 2).

2. Draw FA, cutting the side of the pentagon in G.

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, O.

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.

B

E

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.

3. From C, cut off CF, equal to BE, and draw the chord EF. EF is a side of the required square. Complete the square

(Pr. 34), and EFGH will be the required square, inscribed 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 equilateral figure, inscribed in a given parallelogram ABCD.

Problem 105.

To inscribe a rectangle in a given triangle ABC, having a side equal to a given line DĚ.

1. On BC, mark off BF equal to DE.

2. Through F, draw. FG parallel to AB; and through G, 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.

To inscribe an octagon in a given square ABCD.

1. Draw the diagonals AD, BC, intersecting each other

in E.

2. From A, B, C, and D, with CE as radius, describe quadrants cutting the sides of the square in F, G, H, K, L, M, N, O.

3. Join these points, and the required octagon will be inscribed in the given square ABCD.

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