MYSTERIES OF THE EQUILATERAL TRIANGLE - HIKARI Ltd

Chapter 5
Mathematical Competitions
As should be abundantly clear from the previous chapters, the equilateral
triangle is a fertile source of mathematical material which requires neither elaborate
mathematical technique nor heavy mathematical machinery. As such,
it has provided grist for the mill of mathematical competitions such as the
American Mathematics Competitions (AMC), USA Mathematical Olympiad
(USAMO) and the International Mathematical Olympiad (IMO).
Problem 1 (AMC 1951). An equilateral triangle is drawn with a side of
length of a. A new equilateral triangle is formed by joining the midpoints of
the sides of the first one, and so on forever. Show that the limit of the sum of
the perimeters of all the triangles thus drawn is 6a. [262, p. 12]
Problem 2 (AMC 1952). Show that the ratio of the perimeter of an equilateral
triangle, having an altitude equal to the radius of a circle, to the perimeter
of an equilateral triangle inscribed in the circle is 2 : 3. [262, p. 20]
Figure 5.1: AMC 1964
129