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Why Must There Be 4-way Intersections With Equal Angles?
We'll tackle this question in two steps. First we'll use Euler's Formula to show that there must be 4-way intersections created when great circles are applied to a sphere to create triangles. Then we'll use some basic geometry to show that those angles must all be the same and thus must all be right angles.
Why Are There 4-way Intersections?
Euler's Formula relates the number of edges, faces and vertices of a solid: F + V = E + 2
Even though our triangles are on the surface of a sphere, much of the geometry is similar to that of a polyhedron, since essentially the division is a projection of the polyhedron onto the surface of the sphere, so Euler's Formula applies.
Now, since we have triangles we know that there are three edges for every face, and we know that each edge is shared by two faces thus: E = (3/2)F
Assume that there are more than four triangles (faces) at each vertex. Then there must be at least 6 faces for each vertex. Since each face has three vertices (it's a triangle) we have:
F ≥ (6/3)V
F ≥ 2V
F/2 ≥ V
So, F + F/2 ≥ F + V (using the above inequality)
and F + V = 3/2F + 2 (substituting E = (3/2)F into Euler's Formula)
thus, F + F/2 ≥ (3/2)F +2 which is a contradiction since F + F/2 = (3/2)F.
So, our original assumption that there are more than four triangles at each vertex must be wrong and there is at least one vertex with only four triangles.
Why Must the Angles at a 4-way Intersection Be Equal?
Consider a 4-way intersection. If the angles at the intersection are not equal then they must occur pairwise and opposite of each other.
The sides of the triangle that make up angle α must be in the opposite orientation for the opposite angle α since we have congruent triangles.
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And, since the four triangles at that vertex are congruent, there must be an angle α for one of the other angles of the shaded triangle, so the third side of that triangle must be 1 or 2.
In any case you have an isosceles triangle (whose base angles must be equal) which requires angle α = angle β. Thus, the four angles at the vertex are equal and are 90° angles.