MYSTERIES OF THE EQUILATERAL TRIANGLE - HIKARI Ltd

142 Mathematical Competitions
Figure 5.19: Iberoamerican Mathematical Olympiad (Mexico) 1993
Problem 50 (Iberoamerican Mathematical Olympiad (Mexico) 1993).
Let ABC be an equilateral triangle and Γ its incircle (Figure 5.19). If D and
E are points of the sides AB and AC, respectively, such that DE is tangent
= 1. [306, p. 9]
to Γ, show that AD
DB
+ AE
EC
Figure 5.20: Mathematical Olympiad of the Republic of China 1994
Problem 51 (Mathematical Olympiad of the Republic of China 1994).
Let ABCD be a quadrilateral with AD = BC and let ∠A+∠B = 120 ◦ . Three
equilateral triangles ∆ACP, ∆DCQ and ∆DBR are drawn on AC, DC and
DB, respectively, away from AB (Figure 5.20). Prove that the three new vertices
P, Q and R are collinear. [306, p. 11]