Solution
Nearest Wall of a Triangle
Show the problem again
A point is chosen uniformly at random inside an equilateral triangle of height 1. What is the expected distance from the point to the nearest side of the triangle? Give a reduced fraction.
Worked solution
The answer is 1/9. Let D be the distance to the nearest side. The points at distance at least t from all three sides form a smaller equilateral triangle with the same centre, whose inradius is 1/3 − t (the original inradius is one third of the height). It is a scaled copy with ratio (1/3 − t)/(1/3) = 1 − 3t, so P(D ≥ t) = (1 − 3t)² for 0 ≤ t ≤ 1/3. Then E[D] = ∫₀^{1/3} (1 − 3t)² dt = (1/3)·(1/3) = 1/9. Compare Viviani's theorem: the three distances always sum to exactly 1, so each averages 1/3; the nearest one averages a third of that.
Source: Classic geometric-probability exercise in wide circulation; no traceable original source. Statement written for AxiomIQ.