MYSTERIES OF THE EQUILATERAL TRIANGLE - HIKARI Ltd

130 Mathematical Competitions
Problem 3 (AMC 1964). In Figure 5.1, the radius of the circle is equal to
the altitude of the equilateral triangle ABC. The circle is made to roll along
the side AB, remaining tangent to it at a variable point T and intersecting
sides AC and BC in variable points M and N, respectively. Let n be the
number of degrees in arc MTN. Show that, for all permissible positions of the
circle, n remains constant at 60 ◦ . [263, p. 36]
Problem 4 (AMC 1967). The side of an equilateral triangle is s. A circle
is inscribed in the triangle and a square is inscribed in the circle. Show that
the area of the square is s 2 /6. [264, p. 15]
Problem 5 (AMC 1970). An equilateral triangle and a regular hexagon have
equal perimeters. Show that if the area of the triangle is T then the area of the
hexagon is 3T/2. [264, p. 28]
Figure 5.2: AMC 1974
Problem 6 (AMC 1974). In Figure 5.2, ABCD is a unit square and CMN
is an equilateral triangle. Show that the area of CMN is equal to 2 √ 3 − 3
square units. [11, p. 11]
Problem 7 (AMC 1976). Given an equilateral triangle with side of length
s, consider the locus of all points P in the plane of the triangle such that the
sum of the squares of the distances from P to the vertices of the triangle is a
fixed number a. Show that this locus is the empty set if a < s 2 , a single point
if a = s 2 and a circle if a > s 2 . [11, p. 24]