Problem 70.

To inscribe a regular do-decagon within a given circle A.

1. Draw any two diameters BC and DE at right angles to each other.

2. With centres D, B, E, and C, and the radius of the circle AB, describe arcs cutting the circumference in F, G, H, K, L, M, N, and O.

3. Join DO, OG, &c., by straight lines, and a regular dodecagon will be inscribed within a given circle A.

Problem 71-(A.)

To construct any regular polygon (say a pentagon) on a given straight line AB.

General Method.

1. Produce the side AB, in this case, say towards the left. With A as centre and AB as radius, describe the semicircle.

2. Divide the semicircle into as many equal parts as the polygon is to have sides (five).

3. From A draw A 2, to the second division of the semicircle. This makes another side of the required figure.

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4. Bisect the two sides 2 A, AB, by lines CD, DE, and from centre D, their point of intersection, and radius DA, describe the circumscribing circle.

5. Mark off, on the circumference, the divisions 2 E, EF, equal to AB. Join 2 E, EF, FB, and the pentagon is constructed on the given line AB.

NOTE.-A line must always be drawn from A to the second division on the semicircle, no matter how many sides the polygon is to have.

(B.)

Another General Method.

1. At point B raise a perpendicular BC equal to AB (Pr. 2), and describe the quadrant AC.

2. Divide AC into as many equal parts as the required polygon is to have sides (five).

3. Draw a line from B to the second point of division.

4. Bisect AB in D (Pr. 1), and from D erect a perpendicular to meet B 2 in E. (Pr. 2).

5. From centre E, with EA radius, describe a circle; it will contain the required polygon.

E

6. With AB as radius, starting from A, cut the circle in the points F, G, H successively.

7. Join AF, FG, &c., by straight lines, and a regular pentagon will be constructed upon a given straight line

AB.

Problem 72.

To construct a regular hexagon on a given line AB.

1. With points A and B as centres, and radius AB, describe the arcs intersecting at C.

B

2. From the point C, with CA as radius, describe the circle. 3. From D and E, with the same radius, cut off F and G.

4. Join AD, DF, &c., by straight lines, and ADFGEB is the required hexagon.

Problem 73.

To construct a regular heptagon on a given line AB.

1. From B as centre, with radius AB, describe a semicircle cutting AB produced in C.

2. From A, with the same radius, cut the semicircle in D. 3. Bisect AB in E (Pr. 1), and join DE.

K

H

D

E

B

4. From C, with DE as radius, cut the semicircle in F.

5. Join BF; it is another side of the heptagon.

6. Find the centre of the circle that contains it, and complete the heptagon. Then AB......L will be the heptagon required.

Problem 74.

To construct a regular octagon on a given line AB.

1. Produce AB both ways, and erect perpendiculars at A and B (Pr. 2).

2. From A and B, with radius AB, describe the quadrants CE, FD.

3. Bisect these quadrants in the points G and H respectively.

4. Join AG, BH; these will be two more sides of the octagon.

5. Join GH, and at G and H erect perpendiculars GK, HL, equal to AB (Pr. 2).

6. Join KL, and make the perpendiculars at A and B equal to GH or KL-viz., AM and BN.

7. Join KM, MN, and NL, and the required octagon will be constructed on the given line AB.

Problem 75.

To construct a regular nonagon on a given line AB.

1. Produce the line AB; and from B, with radius BA, describe an arc cutting the produced line AB in C, and being produced below A.

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2. From A, with the same radius, describe an arc, cutting

the first arc in D and E.