reading a book of spherical astronomy I've read this:
Three great circles pass through three points on a sphere. If for each great circle we consider only one of the two parts in which it is divided by the two points that determine it, we will have a spherical triangle. Three points on the sphere thus define eight spherical triangles, one of which is entirely situated in a hemisphere, i.e. such that the three arches that make it up are all smaller than a semicircle.
Now, the "smallest" triangle is obviously clear, but I can't understand exactly (… probably an image could help…) how the other triangles are built, and how an arch – the side of the triangle – can be greater than a semicircle.
Perhaps a mistake in the book or an imprecise description?
Thanks in advance
Carlo
Best Answer
Suppose the three points are $A$, $B$, $C$, and let $P$ be the center of the sphere. Then we have three planes that pass through $P$, namely $ABP$, $ACP$, and $BCP$.
Consider a given point on the sphere; it can be on one side or the other of each of the three planes, so it can lie in one of $2 \times 2 \times 2$ regions of the sphere. These 8 regions are the 8 spherical triangles.
Another explanation: each of the three planes divides the sphere into two pieces. So, the three planes together divide the sphere into $2 \times 2 \times 2$ pieces.