MYSTERIES OF THE EQUILATERAL TRIANGLE - HIKARI Ltd

Mathematical Competitions 135
Figure 5.11: AMC-10 2004
Problem 19 (AMC-10 2004). In Figure 5.11, points E and F are located
on square ABCD so that ∆BEF is equilateral. Show that the ratio of the area
of ∆DEF to that of ∆ABE is 2 : 1. [99, p. 34]
Figure 5.12: AMC-10 2005
Problem 20 (AMC-10 2005). The trefoil shown in Figure 5.12 is constructed
by drawing circular sectors about sides of the congruent equilateral
triangles. Show that if the horizontal base has length 2 then the area of the
. [99, p. 42]
trefoil is 2π
3
Problem 21 (AMC-12 2003a). A square and an equilateral triangle have
the same perimeter. Let A be the area of the circle circumscribed about the
square and B be the area of the circle circumscribed about the triangle. Show
. [323, p. 18]
that A
B
= 27
32