My propositional logic about the topic is following and My question is the way to prove this: If two or more points(vertex) of a 3D polygon share the same solid angle, then those points are on the same sphere.
Please note that solid angles that I mentioned is originated from individual points, not the one common coordinate. For instance, the solid angle of 12th point on the below picture is calculated by three vectors from 12 to 23, 5, and 24.
The polygon on the above picture has 84 points. I highlighted(yellow) points that has the same "solid angle". And I think they are on the same sphere that is smaller in size than the polygon. But I am not sure where to start to prove it.