MYSTERIES OF THE EQUILATERAL TRIANGLE - HIKARI Ltd

136 Mathematical Competitions
Problem 22 (AMC-12 2003b). A point P is chosen at random in the interior
of an equilateral triangle ABC. Show that the probability that ∆ABP
. [323, p. 20]
has a greater area than each of ∆ACP and ∆BCP is equal to 1
3
Problem 23 (AMC-12 2005). All the vertices of an equilateral triangle lie
on the parabola y = x 2 , and one of its sides has a slope of 2. The x-coordinates
of the three vertices has a sum of m/n, where m and n are relatively prime
positive integers. Show that m + n = 14. [323, p. 46]
Problem 24 (AMC-12 2007a). Point P is inside equilateral ∆ABC. Points
Q, R and S are the feet of the perpendiculars from P to AB, BC and CA,
respectively. Given that PQ = 1, PR = 2 and PS = 3, show that AB = 4 √ 3.
[323, p. 63]
Problem 25 (AMC-12 2007b). Two particles move along the edges of equilateral
∆ABC in the direction A → B → C → A, starting simultaneously and
moving at the same speed. One starts at A and the other starts at the midpoint
of BC. The midpoint of the line segment joining the two particles traces out a
path that encloses a region R. Show that the ratio of the area of R to the area
of ∆ABCis 1 : 16. [323, p. 64]
Figure 5.13: USAMO 1974
Problem 26 (USAMO 1974). Consider the two triangles ∆ABC and ∆PQR
shown in Figure 5.13. In ∆ABC, ∠ADB = ∠BDC = ∠CDA = 120 ◦ . Prove
that x = u + v + w. [196, p. 3]