Mathematics
Problem Of the Week
Spring
2002
POW #10
Active/Passive Triangles
Define the center of a triangle as the unique intersection point of the
three medians drawn from vertices to bisect the opposite sides. Label the center point as
"passive" and label each of the vertices as "active". A point in the triangle will be called "active"
if it is closer to a vertex than it is to the center and otherwise we'll call
it "passive". We are
interested in what proportion of the total area of the triangle is active.
Will this ratio be the same for all triangles? To investigate this question, you might want
to consider some special cases (e.g., equilateral triangles, right triangles, isosceles
triangles).
The most general solution will take this week's prize.