The "inscribed angle theorem" is a common 2-dimensional plane geometry fact. It states that for a circle the angle formed between any two points on the circumference with the center is twice the angle formed by those two points with any other point on the circumference. I will not elaborate on a proof or further details here, but instead provide a link and image from the wikipedia page on this topic, where the basic theorem is proved.
My question is whether this simple 2D geometric concept can be adapted to solid angles in 3D? And perhaps beyond to n dimensions?
The 2D case dealt with 3 points on the circumference of a circle (2 that defined the arc, and the 3rd point that formed the angle that was half the angle at the center). In 3D, imagine a sphere rather than a circle, and consider 4 points instead of 3. Let 3 of the 4 points form the base of a tetrahedron, then consider two distinct cases.
In the first case the 4th point forms the tip of the tetrahedron. In the other case the center of the sphere defined by the 4 points forms the tip of the tetrahedron. In either case the base of the tetrahedron, and the associated spherical triangle on the surface of the sphere, is the same. However there are two different solid angles at the tip of the tetrahedron in each case -- one for when the tip is at the center of the sphere and the other when the tip is at the 4th point defining the sphere.
Are these solid angles related (one being half of the other, or some other similar relation) as in the inscribed angle theorem in 2D? If they are related in some way, is this a common fact/theorem in solid geometry? Does it have a name like the "inscribed angle theorem" in 2D? What is the relation between these two solid angles?
Is there a similar concept in 4D, or n-dimensional, space? (I am not even sure if there is a solid angle concept in arbitrary n-dimensional space.) If there is a concept of solid angles in higher dimensions is there a predictable relation between these two angles a set n-dimensional space?For example, a particular dimension like 10-dimensional space, then 11 points in that 10D space, is there a way to easily find the "solid angle" at the 11th point, and then use a fixed relation to know the "solid angle" at the center of the 10D hypersphere going out to the other 10 points on the surface of the hypersphere?