We are given that the equidistant set of two points on $\mathbb{S^2}$ is a "line" (great circle) on $\mathbb{S^2}$. Using this fact I must show that:
If $P,Q \in \mathbb{S^2}$ are equidistant from $A,B,C \in\mathbb{S^2}$, the $P=Q$. Deduce from this result that an isometry of $\mathbb{S^2}$ is determined by the images of three points $A,B,C$ not in a "line".
My work:
"the equidistant set of two points on $\mathbb{S^2}$ is a "line" (great circle) on $\mathbb{S^2}$ "
I am able to understand this result. My understanding is that the reason that this is true is because any two points on the sphere are simply points in $\mathbb{R^3}$ that are a distance of $1$ unit from the origin, and the set of points equidistant to this pair of points forms a plane in $\mathbb{R^3}$ that passes through the origin. In $\mathbb{S^2}$, this is simply the intersection of this plane with $\mathbb{S^2}$, so the set of points equidistant to the given pair is a great circle or "line" on $\mathbb{S^2}$. But how do I use this to answer the above question?
I am having a hard time visualizing this, and I started out by just writing equations of distance between $P$ and $Q$ and $A,B,C$ and trying to equate them but that gets confusing and messy fast and i'm not sure if it's going in the right direction.
Another idea that I had was first considering the set of equidistant points between $A$ and $B$, which would be a great circle, which we will call $I$. Then consider the set of equidistant points between $A$ and $C$, which would be a great circle, which we will call $J$. Finally, we consider the set of equidistant points between $B$ and $C$, which would again be a great circle, which we will call $K$. Then the points at the intersection of the great circles $I,J,K$ would be equidistant to $A,B,C$. Intuitively I think (I don't know this for sure), these three great circles would intersect in two points. One point would be in the 'triangle' (not sure what to call this, but I mean the curved triangle) $ABC$, and one would be outside $ABC$ on the other side of the sphere. But this would mean that that there are two unique points that are equidistant from $A,B,C$, so i'm not sure if this line of thinking is correct either.
As for the second part about the isometries of $\mathbb{S^2}$, I have no idea where to even begin. Any help would be great and I would really appreciate if it is dumbed down a little bit because I am a complete beginner at non-Euclidean geometry. Thank you!
The set of points equidistant from $A$ and $B$ can be visualized as a (tilted) equator with $A$ and $B$ symmetrical like approximately Porto Rico (Northern H.) and Bogota (Southern H.) are with respect to our usual equator. Let us call this circle $M_{AB}$ ($M$ like "mediator").
You have seen where the problem is : the issue is that the intersection of the 3 "mediators" $M_{AB}, M_{BC}, M_{CA}$ is not a single point, equivalent for a spherical triangle to the circumcenter for a plane triangle, but 2 antipodal points.