In the comments [J. M.][1] linked to a [Demonstration][2] by Borut Levart.

Here is code from that, with my own refactoring:

    eps = 10^-6;

    (* from spherical to cartesian coordinates *)
    sp[{ϕ_, θ_}] := {Sin[θ]*Cos[ϕ], Sin[θ]*Sin[ϕ], Cos[θ]}
    
    (* part of great circle between two sphere points *)
    ark[{r1_, r2_}, nt_] := Table[
      RotationTransform[t VectorAngle[r1, r2],
        Cross[r1, r2]][r1], {t, 0, 1, 1/nt}]
    
    Manipulate[
     If[p1 == p2, p1 = .99 p2];
     If[p1 == p3, p1 = .99 p3];
     If[p3 == p2, p3 = .98 p2];
     Graphics3D[{
       {Red, Opacity[.6], Sphere[{0, 0, 0}, .995]},
       ark[#, 20] & /@ Subsets[sp /@ {p1, p2, p3}, {2}] // Line,
       PointSize[.015], Point[sp /@ {p1, p2, p3}]
      },
      Boxed -> False,
      ImageSize -> {400, 400},
      FaceGrids -> {{0, 0, -1}},
      FaceGridsStyle -> GrayLevel[.5]
     ],
     {{p1, {4.2, .5}, "point one"}, {eps, π (1 - eps)}, {2 π (1 - eps), eps}},
     {{p2, {.1, 1.1}, "point two"}, {eps, π (1 - eps)}, {2 π (1 - eps), eps}},
     {{p3, {5.1, 1.8}, "point three"}, {eps, π (1 - eps)}, {2 π (1 - eps), eps}},
     ControlPlacement -> Left, SaveDefinitions -> True
    ]

![enter image description here][3]


  [1]: http://mathematica.stackexchange.com/users/50/j-m
  [2]: http://demonstrations.wolfram.com/TrianglesOnASphere/
  [3]: https://i.sstatic.net/pHRfQ.png