Let $u,v,w \in S^2$ and let $[u,v,w]$ be a spherical triangle. I want to find its inscribed circle. However, I don't know how to approach this problem. Would someone please explain?
In my understanding, one needs to somehow find the altitudes of the spherical triangle and then the point of its intersection (circumcentre)? Then what about the radius?
If $o$ is the center of the sphere, consider planes $ouv$, $ovw$ and $owu$. The bisector planes of the angles, formed by any two of those planes, meet at a common line $r$, having the same distance from the planes.
You can then construct the plane passing by line $r$ and perpendicular to plane $ouv$: that plane cuts arc $uv$ at some point $p$. The circle having $r$ as perpendicular axis and passing through $p$ is the requested incircle.