1-2-3-circle triangles reduced by reflection and rotation

Author:
String_1
Topic:
Area
What it the expected area of a triangle formed by random points on circles with radii 1, 2 and 3 respectively?
Try changing alpha and beta by moving the points on the line at the bottom. Note how S1 and S2 defines the positions of P3 where the formula BB=sin(alpha)+3sin(beta)-1.5sin(alpha+beta) changes sign. S2 only works for alpha<pi.