23. To Inscribe an Equilateral Triangle within a Given Circle. - In Fig. 153, let A B D be any given circle within which an equilateral triangle is to be drawn. From any point in the circumference, as E, with a radius equal to the radius of the circle, describe the arc D C B, catting the given circle in the points D and B. Draw the line D B, which will be one side of the required triangle. From D or B as center, and with D B as radius, cut the circumference of the given circle, as shown at A. Draw A B and A D, which will complete the figure.

24. To Inscribe a Square within a Given Circle-In Fig. 154, let A C B D be any given circle within which it is required to draw a square. Draw any two diameters at right angles with each other, as C D and A B. Join the points C B, B D, D A and A C, which will complete the required figure.

Fig. 155 - To Inscribe a Regular Pentagon within a Given Circle.

Fig. 150. - To Inscribe a Regular Hexagon within a Given Circle.

Fig. 157. - To Inscribe a Regular Heptagon within a Given Circle.

25. To Inscribe a Regular Pentagon within a Given Circle. - In Fig. 155, ADBC represents a circle in which it is required to draw a regular pentagon. Draw-any two diameters at right angles to each other, as A B and D C. Bisect the radius A II, as shown at E. With E as center and E D as radius strike the arc D F, and with the chord D F as radius, from D as center, strike the arc F G, cutting the circumference of the given circle at the point G. Draw D G, which will equal one side of the required figure. With the dividers set equal to D G, step off the spaces in the circumference of the circle, as shown by the points I K and L. Draw D I, I K, K L and L G, thus completing the figure.

26. To Inscribe a Regular Hexagon within a Given

Circle. - In Fig. 156, let A B D E F G be any given circle within which a hexagon is to be drawn. From any point in the circumference of the circle, as at A, with a radius equal to the radius of the circle, describe the arc C B, cutting the circumference of the circle in the point B. Connect the points A and B. Then A B will be one side of the hexagon. With the dividers set to the distance A B, step off in the circumference of the circle the points G, F, E and D. Draw the connecting lines A G. G F, F E, E D and D B, thus completing the figure. By inspection of this figure it will be noticed that the radius of a circle is equal to one side of the regular hexagon which may be inscribed within it. Therefore set the dividers to the radius of a circle and step around the circumference, connecting the points thus obtained.

27. To Inscribe a Regular Heptagon within a Given Circle. - In Fig. 157, let F A G B H I K L D be the given circle. From any point, A, in the circumference, with a radius equal to the radius of the circle, describe the arc BCD, cutting the circumference of the circle in the points B and D. Draw the chord B D. Bisect the chord B D, as shown at E. With D as center, and with D E as radius, strike the arc E F, cutting the circumference in the point F. Draw D F, which will be one side of the heptagon. With the dividers set to the distance D F, set off in the circumference of the circle the points G H I K and L, and draw the connecting lines F G, G H, H I, I K, K L and L D, thus completing the figure.

28. To Inscribe a Regular Octagon within a Given Circle. - In Fig. 158, let B I D F A G E H be the given circle within which an octagon is to be drawn. Draw any two diameters at right angles to each other, as B A and D E. Draw the chorda D A and A E. Bisect D A, as shown, and draw L H. Bisect A E and draw K I. Then connect the several points in the circumference thus obtained by drawing the lines D I, I B, B H, H B, E G, G A, A F and F D, which will complete the figure.

29. To Inscribe a Regular Nonagon within a Given Circle. - In Fig. 169, let M K F E be the given circle.

Draw any two radii at right angles to each other, as B C and A C, and draw the chord B A. From A as center, and with a radius equal to one-half the chord A B, as shown by A D, strike the arc D E, cutting the circumference of the circle at the point E. Draw A E, which will be one side of the nonagon. Set the dividers to the distance A E and step off the points M, H, K, G, I, F and L, and draw the connecting lines, as shown, thus completing the figure.

Fig. 158. - To Inscribe a Regular Octagon within a Given Circle.

Fig. 159. - To Inscribe a Regular Nonagon within a Given Circle.

Fig. 160. - To Inscribe a Regular Deca-gon within a Given Circlc.

30. To Inscribe a Regular Decagon within a Given Circle.-In Fig. 160, let DBE A be any given circle in which a decagon is to be drawn. Draw any two diameters through the circle at right angles to each other, as shown by B A and D E. bisect b C, as shown ats F, and draw F D. With F as center, and F D as radius, describe the arc DG, cutting B A in the point G. Draw the chord D G. With D as center, and D G as radius, strike the arc. G II, cutting the circumference in the point H. Connect D and H, as shown. Bisect D H and draw the line C R, cutting the circumference in the point I. Draw the lines H I and I. D, which will then he two sides of the required figure. Set the dividers to the distance H I and space oil' the circumference of the circle, as shown, and draw the connecting lines D K, K M, M L, L P, P E, E N, N O and O H, thus completing the figure.