Problem:
A point is chosen at random in the interior of equilateral triangle . What is the probability that has a greater area than each of and ?
Answer Choices:
A.
B.
C.
D.
E.
Solution:
Since the three triangles , and have equal bases, their areas are proportional to the lengths of their altitudes.
Let be the centroid of , and draw medians and . Any point above will be farther from than from , and any point above will be farther from than from . Therefore the condition of the problem is met if and only if is inside quadrilateral .
.jpg)
If is extended to on , then is divided into six congruent triangles, of which two comprise quadrilateral . Thus has onethird the area of , so the required probability is .
By symmetry, each of , and is largest with the same probability, so the probability must be for each.
The problems on this page are the property of the MAA's American Mathematics Competitions