MYSTERIES OF THE EQUILATERAL TRIANGLE - HIKARI Ltd

146 Mathematical Competitions
Figure 5.27: Miscellaneous #1
Problem 59 (Miscellaneous #1). Altitude AD of equilateral ∆ABC is a
diameter which intersects AB and AC at E and F, respectively, as in Figure
5.27. Show that the ratio EF : BC = 3 : 4 and that the ratio EB : BD = 1 : 2.
[246, p. 17]
Figure 5.28: Miscellaneous #2
Problem 60 (Miscellaneous #2). Show that the ratio between the area of
a square inscribed in a circle and an equilateral triangle circumscribed about
the same circle is equal to 2√ 3
9 (Figure 5.28). Also, show that the ratio between
the area of a square cirmcumscribed about a circle and an equilateral triangle
inscribed in the same circle is equal to 16√ 3
9 . [246, p. 24]