Understanding triangles on 2-sphere

Question

Suppose that you are two-dimensional being, living on the surface of a sphere 
of radius R. Consider an equilateral triangle with the vertices on three 
meridians, each 120 degrees apart and same distance r (r < πR/2) from the North 
pole.

I am trying to understand this statement but I am kind of confused. How can we form a triangle if the all the vertices have the same distance from the north pole ? Shouldnt they form circle rather then a triangle. I think if r wasnt constant then we could have form a triangle. Can someone help me to understand the picture of this triangle.

Answer 1

For some $r$ you could choose:

• 51° 30' N 0° W (London, UK)
• 51° 30' N 120° E (somewhere next to the border between China and Russia)
• 51° 30' N 120° W (Dunn Peak Provincial Park, Canada)

and you get this triangle.

Edit:

Shouldnt they form circle rather then a triangle.

In spherical geometry the “lines” (in the sense of shortest paths) are the great circles. But the 3 points do not lie on a common great circle for $r< \frac{\pi R}{2}$.

For $r = \frac{\pi R}{2}$ the vertices would lie on the equator and the spherical triangle would degenerate to a great circle.

Answer 2

The sides of a spherical triangle lie along great circles. Except for the equator, no line of latitude is a great circle.

The midpoint of each side of the triangle will be closer to the north pole than any of the vertices are.