MYSTERIES OF THE EQUILATERAL TRIANGLE - HIKARI Ltd

134 Mathematical Competitions
Figure 5.9: AMC 1992
Problem 15 (AMC 1992). In Figure 5.9, five equilateral triangles, each with
side 2 √ 3, are arranged so they are all on the same side of a line containing
one side of each. Along this line, the midpoint of the base of one triangle is a
vertex of the next. Show that the area of the region of the plane that is covered
by the union of the five triangular regions is equal to 12 √ 3. [270, p. 26]
Figure 5.10: AMC 1995
Problem 16 (AMC 1995). In Figure 5.10, equilateral triangle DEF is inscribed
in equilateral triangle ABC with DE ⊥ BC. Show that the ratio of the
area of ∆DEF to the area of ∆ABC is 1 : 3. [251, p. 5]
Problem 17 (AMC 1998). A regular hexagon and an equilateral triangle
have equal areas. Show that the ratio of the length of a side of the triangle to
the length of a side of the hexagon is √ 6 : 1. [251, p. 27]
Problem 18 (AMC-10 2003). The number of inches in the perimeter of
an equilateral triangle equals the number of square inches in the area of its
circumscribed circle. Show that the radius of the circle is 3 √ 3/π. [99, p. 23]