Fig. 25
Join the latter points to produce the required hexagon without the given circle. (3) Join the points G, E, A. This will obtain the points 1, 2, 3 on the diameters. Draw 1, 4 perpendicular to G B. With 1, 4 as radius and 1 as centre describe one of the required circles. 3 and 2 are the centres of the other two required circles.
[Fig. 27] Within a given circle to inscribe any number of equal circles, each touching the circumference and two other circles.
Divide the circle in the same number of parts as the number of circles required—in this case five. Draw the five radii. Bisect the angles B D A and A D C. Draw E F perpendicular to D A. D E F is a triangle any two angles of which bisect as at D and E. From point 1 thus obtained on D A and radius 1 A inscribe a circle. From D as centre and D 1 as radius describe a circle cutting the five radii in points 1, 2, 3, 4, 5. With the latter points as centres and 1 A as radius describe the remaining required circles.
[Fig. 28] This problem is worked in the same manner as [Fig. 27], seven circles being inscribed instead of five in a given circle.
[Fig. 29] To inscribe a trefoil, or three equal semicircles having adjacent diameters in a given circle.
Divide the given circle into six equal parts by marking off the length of the radius six times on the circumference. From the centre D to these six points draw radii. Bisect any of the six sectors as at E. Draw E C obtaining F on one of the radials. On either side of F draw lines from it to meet the alternate radials perpendicular to B D and D C, and
