Join the centres E and F. Bisect E F in G, and describe a semicircle on E F. From K on the larger circle mark off K J and E F equal to the radius of the smaller circle, and with F as centre and F J as radius describe an arc passing through semicircle at H. Join F H cutting the larger circle at C, and draw E A parallel to F H. The points of contact are A and C, through which the interior tangent is drawn.
[Fig. 17] Within a given circle to describe any Regular Polygon—say a Pentagon.
Draw the diameter A F and divide it into the same number of parts as the required polygon is to have sides—in this case it will be five parts. To divide the diameter into the number of equal parts, draw a line A X any angle to A F. Set off any convenient measurement five times on this line. Join point 5 to F, and draw the lines 4, 4´, 3, 3´, &c., parallel to 5 F to meet the diameter. With A and F as centre and A F as radius describe arcs intersecting at L. From
Fig. 15
L draw a line through the Second division on A F at point 2´ cutting the circumference at B. Join A B. This is the length of the side of the required polygon. Set off the length of the side A B around the circumference at C, D, and E. Join the points A, B, C, D, E to complete the required pentagon.
N.B.—A Regular Hexagon may be inscribed in a circle by setting off the length of its radius six times round the circumference, and joining the points.