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1/4 is the answer. Use angles. Fix the first point arbitrarily. look at the average distance between 2 random points.
Its hard to explain without a diagram, and i don't have web hosting.
ok. fix point A. consider half of the square, starting from point A to the reflection of point A through the center, O, which we shall call a.
randomly generate point B. let b be the reflection of B through the origin. B obviously lies between A and a.
Let AB be the shorter line segment connection A and B.
now, in order for the triangle to cover the center, O, point C has to fall between a and b. that is, on the line segment ab.
note that ab has the same length as AB.
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