MYSTERIES OF THE EQUILATERAL TRIANGLE - HIKARI Ltd

Mathematical Properties 47
Figure 2.23: Abutting Equilateral Triangles [179]
Property 31 (Abutting Equilateral Triangles). Equilateral triangles of
sides 1, 3, 5, . . . , 2n − 1, . . . are placed end-to-end along a straight line
(Figure 2.23). The vertices which do not lie on the line all lie on a parabola
and their focal radii are all integers [179].
Figure 2.24: Circumscribing Rectangle [181]
Property 32 (Circumscribing Rectangle). Around any equilateral triangle
ABC, circumscribe a rectangle PBQR (Figure 2.24). In general, each side of
ABC cuts off a right triangle from the rectangle. The areas of the two smaller
right triangles always add up to the area of the largest one (X = Y +Z) [181].