Proof of EAT in Spherical Geometry
The proof of EAT fails on some triangles in spherical geometry. See the proof construction in spherical geometry above. (The complete written proof is omitted here.)
In the proof construction, we can make the triangle ABC large enough to make B-E-B'-F, where B' is the intersection of spherical lines BE and BC, which is not B (i.e., the antipodal point of B). Then we can see that angle FCA is greater than angle DCA. This shows the interior angle CAB is greater than the exterior angle DCA, since the angle FCA and CAB are congruent.
Go back to the proof construction in spherical above. Try to stretch triangle ABC (e.g., dragging A up and down) and see how B' and F on the backside change and how angles FCA and DCA change.
This is possible in spherical geometry, but not in Euclidean geometry because the proof construction includes spherical lines BE and BC meeting at two distinct points B and B', which would violate the line postulate in Euclidean; two distinct points determine one and only one line (alternatively, if two lines meet at two distinct points, they are identical to each other). The only way to make angle FCA close to DCA in Euclidean geometry is to drag A down so that triangle ABC collapses, as B, A, and C look collinear.
Go to the proof construction in Euclidean below. Try to make FCA closer to DCA. When do you see that FCA and DCA are congruent?
Thus, in Euclidean geometry, the proof works for every triangle, so the EAT holds in general. On the other hand, in spherical geometry, the proof works for triangles where B-E-F-B' so does EAT hold for them. But, when F=B' or B-D-B'-F, then it fails.
Then, when does EAT hold in spherical geometry, if it does not hold for all triangles but some?
We can characterize when the EAT holds or not by looking at the triangle's median. When F=B' and DCA and FCA are congruent, the segment BF (or BB') is exactly half of the great circle, and the median of triangle ABC, BE, is exactly 1/4 of the great circle. When B-D-B'-F and DCA < FCA, BE is greater than 1/4 of the great circle.
Thus, when a triangle in spherical geometry has its median greater than or equal to 1/4 of the great circle, the proof fails, so that the EAT does not hold. To be specific, the exterior angle at one vertex of the side where the median bisects (e.g., DCE at C) is smaller than the interior angle at the other vertex of the side (e.g., CAB at A).
So, we can modify the statement of EAT in spherical geometry by restricting the scope of triangles to those whose medians are smaller than 1/4 of the great circle.