Fig. 21
and O 1 as radius, describe a circle to obtain the points 1, 2, 3, 4. These are the centres of the required circles.
N.B.—If the central portion made by the meeting of the four circles were removed, the remaining parts of the circles would form a figure known as the quatrefoil, a form common in architecture.
[Fig. 25] To inscribe six equal circles in a given equilateral triangle A B C.
Bisect the angles of the given equilateral triangle as at E, and draw the bisection lines through to meet the centre of each side. Bisect the angle A B J to obtain the point D on C K. Through D draw G F parallel to A B, also F H and H G parallel to the sides of the triangle. With D as centre and D K as radius inscribe one of the required circles, and with the same radius and F, 2, H, 1, and G as centres inscribe the remaining circles.
[Fig. 26] (1) Within a given circle to inscribe a hexagon. (2) Without the same circle to describe a hexagon. (3) Within the inner hexagon to inscribe three equal circles each touching each other and two sides of the hexagon.
(1) Mark off the length of the radius of the given circle B D F six times on the circumference as at D E F, &c. Draw the three diameters A D, B E, and G F, and produce them a little beyond these points. Join the points G, D, E, F, &c., by straight lines to produce the hexagon within the given circle. (2) Bisect the angle K O H, the line of bisection will cut the circle at point R. Through R draw H K parallel to B C. With O as centre and O H as radius describe a circle cutting the produced diameters at K, L, M, &c.
