Let the three points A,B,C be the vertices of a moving spherical triangle on the surface of a sphere. The triangle moves so that while the vertices A,B remain fixed, the angle BCA at the vertex C stays constant. What is the locus of the moving vertex C? Is there a special name for the curve traced out by C? If A,B,C were the vertices of a plane triangle, the corresponding locus would be the arc of a circle. I have made some calculations, which-if they do not contain errors-lead me to believe that, in the spherical case, the locus I am seeking is not the arc of a small circle (on the sphere). But, if so, I do not know what type of curve it is.

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feaguelaBapzo · Accepted Answer

Let O be the centre of the sphere. Then the spherical BCA angle   γ is the angle between the planes through O,A,C and O,B,C. So we can find our locus in the following way: we take a plane   π through O,A and find a plane   τ through O,B such that the angle between   π and   τ equals   γ. By intersecting   π  ∩  τ with the sphere we find every point of our locus. As an alternative, we may exploit the spherical cosine rule:                    (1)                    cos        ⁡        C        =                              cos            ⁡            c            −            cos            ⁡            a            cos            ⁡            b                                sin            ⁡            a            sin            ⁡            b                              where   c  =                    A        O        B            ^       and so on. Notice that both   c and   C are fixed, so everything boils down to a not-so-trivial constraint between   a and   b.

Let the three points A,B,C be the vertices of a moving spherical triangle on the surface of a sphere

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