3. Make angle FOK equal to angle ABD, and angle FOL equal to angle ACE (Pr. 10).

GM

PH

4. Through K, draw MN at right angles to KO, and through L, draw NP at right angles to LO (Pr. 2); then the triangle MNH is described about the given circle O, and has its angles equal to those of the given triangle ABC.

Problem 132.

To describe a pentagon about a given pentagon ABCDE, having its sides parallel to it, and equal to a given straight line FG.

E

F

1. Find the centre of the pentagon by bisecting two

adjacent angles (Pr. 4).

2. From the centre O, draw the five radii, and produce them indefinitely.

3. Produce one of the sides, as AB, until it is equal to FG; viz. AH.

4. From H, draw a line parallel to OA, cutting the radius OB in K.

5. From 0, with radius OK, describe a circle cutting the
produced radii in L, M, N, P.

6. Join KL, LM, &c., by straight lines, and a pentagon
KLMNP will be described about a given pentagon
ABCDE, and having its sides equal to the given line
FG.

Problem 133.

To describe a square about a given circle A.

1. Draw the diameter BC. With centres B and C, and any radius, describe arcs at D. From D, draw the diameter EF.

2. With centres B, E, C, and F, and radius BA, describe arcs cutting at G, H, K, L.

3. Join GH, HL, LK, and KG, and GHLK is the required square described about the given circle A.

Problem 134.

To construct any regular polygon (say a hexagon) about a given circle A.

1. Divide the circumference into as many equal parts as the polygon is to have sides—six (Pr. 64).

2. Draw radii to these points of division, and produce them beyond the circumference.

3. Join 1 2, and draw BC parallel to 1 2, and tangential to the circle (Pr. 54).

4. Take the distance from the centre of the circle to C, and mark off from the centre points D, E, &c.

5. Join CD, DE, &c., by straight lines, and a regular hexagon BCDEFG will be constructed about a given circle A.

Problem 135.

To describe a circle about a given triangle ABC.

1. Bisect any two of its sides AB, AC by lines cutting in D

(Pr. 1).

2. From D as centre, with DA, DB, or DC as radius, describe a circle; then ABC is the required circle, described about the given triangle ABC.

To describe a circle about a given square ABCD.

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

B

2. From E as centre, with radius EA, draw a circle ABDC, which will be described about the given square ABCD.

Problem 137.

To describe six equal circles about, and equal to, a given circle A, touching each other and the given circle.

1. From the centre A of the given circle, and with its diameter as radius, describe the circle BCDEFG.

2. Draw the diameter BAE, and from B, with the radius of the given circle, describe a circle touching it.

3. From B, mark off the other centres C, D, E, &c.

4. From these points C, D, E, &c., with the radius of the given circle, describe the remaining five circles, which will touch each other, and the given circle A.

Problem 138.

To describe any number of equal circles about a given circle A, each touching two others, and the given circle. (Say eight in this case). 1. Divide the circumference into eight equal parts, and draw produced radii through the points of division.

2. Bisect one of the angles at the centre, as 142 (Pr. 4), by a line AB, and draw a tangent CD at point 1 (Pr. 54), cutting the bisecting line in D.

3. Bisect the obtuse angle CDB (Pr. 4) by a line, which, produced, cuts the radius A1 produced in E.

4. From A as centre, with radius AE, describe the outer circle, cutting the produced radii in F, G, H, &c.