Thales semicircle theorem says that an angle inscribed in a semicircle is a right angle.

Q1. Does a cone with apex on a hemisphere and encompassing the circular base have a solid angle independent of the position of the apex?

It appears it might be true, with solid angle $(2-\sqrt{2})\pi \approx 0.59 \pi$ steradians:

Q2. I am seeking a proof or reference for the generalization to $d$ dimensions.

I have not found a reference, which makes me wonder if the answer to Q1 might be No...