Fig. 27
join their extremities, thus making the equilateral triangle 1, 2, 3. On the sides of this triangle describe the three semicircles required by using points 1, 2, and 3 as centres, and 2 F as radius. The completed figure is the trefoil, and the inscribed three semicircles have their diameters adjacent.
[Fig. 30] To describe an equilateral triangle within and without a given circle.
Fig. 30
Draw six radii dividing the given circle into six equal parts. Join their alternate extremities as at L M N. This makes the required equilateral triangle within the circle. Draw tangents to the circle at L M and N, or lines at right angles to L O, M O, and N O. Produce the latter radii to meet the tangents at A B C. A B C is the equilateral triangle without the circle.
N.B.—It will be seen that the triangle B A C is made up of four similar triangles each equal to L M N. Also, if six of the smaller triangles, as A L M, were placed around points A B and C a hexagon would be formed. This figure is very useful in designing geometrical and other repeating all over patterns in ornament.