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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?

Chung. J
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2 Answers2

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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.

Community
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AakashM
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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.

evil999man
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  • It is not the answer I expected. For any three points A, B, C in a sphere. Choose P in the sphere, consider AP, BP, CP and moving P in the sphere, there is some invariant quantity? – Chung. J May 12 '14 at 05:08
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    @Chung.J Edit your post to add that. Currently, it seems you are asking this only. – evil999man May 12 '14 at 05:16
  • What is included in "similar"? Another dissimilar example is, if you project a right angled triangle on sphere from North Pole onto a plane at South Pole you get a bigger similar triangle.(Stereo-graphic projection). – Narasimham Mar 26 '16 at 19:31