+400 Let PQR be an equilateral triangle, centered at O. A point X is chosen at random inside the triangle. Find the probability that X is closer to O than to any of the sides. (In other words, find the probability that XO is shorter than XA, XB, and XC.)
+129852 See the following
I've labeled the center of equilateral triangle XYZ as A
Any point inside equilateral triangle DEF will be closer to A than to any of the sides of XYZ
Each side DEF is 1/2 the length of a side of XYZ.....so the area of DEF = (1/2)^2 [XYZ] = (1/4) [XYZ]
So the probability is (1/4)
