That’s interesting. How did you come up with the idea that it is the average of these numbers? My idea is like this. Consider a 3D coordinate system with a-axis, b-axis and c-axis. Then the criteria that a+b+c=1 and a,b,c being non-negative are represented by a planar triangle (the plane a+b+c=1 have cut the axes at (1,0,0), (0,1,0) and (0,0,1), and these are the three vertices of the triangle mentioned). We call this triangular area A.

The, a triangle can be formed iff no side is greater than or equal to 1/2. This is represented by the triangular area defined by the three mid-points of area A. This smaller area has one-fourth of area A. So the answer is 1/4.