Triangle XYZ is equilateral, with O as its center. A point P is chosen at random. Find the probability that P is closer to point O than to any of the side lengths.
Construct a triangle ABC inside triangle XYZ with O also as its center. Any point inside ABC will be closer to O than a side length of XYZ, so we can find the probability by finding the ratio of [ABC] to [XYZ]. The distance from O to a side of ABC will be half the distance from O to a side of XYZ, so the altitude of ABC will be half of that of XYZ. Because XYZ and ABC are equilateral, the ratio of their side lengths will also be 1:2. The ratio of the area of ABC to the are of XYZ will be (1/2)^2, so the probability is 1/4.