Properties of sphere

Question

Let $C$ be a circle with diameter $\overline{AB}$. Then it is well known that for any $P$ on the circle $C$ the angle $\angle APB =\frac \pi 2$. There are similar results for sphere?

Answer 1

Yes, you can pass a plane passing through $A,B,P$ as any three points are coplanar. You are left with a circle on this plane. You can now conclude.

Answer 2

You might be asking:

Does a cone with apex on a hemisphere and encompassing the circular base have a solid angle independent of the position of the apex?

in which case the good folk over at MathOverflow have done the work for you and proved that the answer is No.

If you are asking about any similar result, though, they have an open question on that.