[Fig. 18] On a given line to construct any Regular Polygon,—say a Pentagon.
Produce the given line A B to R, and with B as centre and A B as radius describe a semicircle A C R. Divide the semicircle into as many parts as the polygon is to have sides—in this case five. Draw a line from point B to the second division point Q C. Bisect A B and B C to find P, which will be the centre of a circle passing through the points A B C. Mark off the points D and E, making the distances C D, D E, and E A each equal to A B. Join C D, D E, and E A to complete the required polygon.
[Fig. 19] Special method of drawing an Octagon in a given circle.
Draw two diameters B F and H D at right angles to each other. Bisect angles H K B and B K D in the lines K A and K C. Produce the lines K A, K C, to meet the circumference at G and E. The eight points thus found on the circumference are joined to make the required octagon.
[Fig. 20] To inscribe an Octagon in a given square.
With each corner of the square as centres, and half the diagonal of the square as radius, describe arcs
Fig. 18
cutting the sides of the square at F, G, H, K, &c. Join these points to complete the required octagon.