Why Are Triangles Always Created When You Apply Great Circles On A Sphere?

Euler's formula relates the number of verticies, edges and faces for a polyhedron. It also applies to this case of placing great circles on the sphere. Here, the intersections of the circles correspond to the verticies (V), the shapes formed are the faces (F) and the segments of the circles between intersections are the edges (E). We can use some basic deductions about the shapes and the number of edges they must have to arrive at a contradiction of Euler, thereby proving the existance of triangles.

Euler's Formula: F + V = E + 2

We've already considered the case where there are only lunes (which has only 2 vertices, the North and South poles), so we will assume there are more than 2 vertices on the sphere. That means that the lunes that are created when there are only two vertices are cut by any great circle not sharing their intersecting points. So, all of the resulting regions are not lunes and have more than 2 edges.

There are at least 4 edges at each vertex (because great circles crossing will create at least 4 edges), and each edge is shared by two vertices. Thus, (4/2)V ≤ E or simplified: 2V ≤ E

We'll assume that there are no triangles for the sake of contradiction. So each face must have at least 4 edges. And, each edge is shared by 2 faces. (4/2)F ≤ E or simplified: 2F ≤ E

Combining these two results we get: 2F + 2V ≤ 2E or F + V ≤ E

But Euler's Formula states that F + V = E + 2 implying that F + V > E so we have a contradiction. So there must be a face with fewer than 4 edges (and more than 2 or it would be a lune). Thus at least one face has three edges and is a triangle.