A great circle on a sphere

Question

The following proposition is from 'Spherical Geometry and Its Applications' by Marshall A. Whittlesey:

Proposition 5.6 If two distinct points on a sphere are not antipodal then there exists a unique great circle passing through them [1]

Let there be two non-antipodal distinct points on a small circle of a sphere. I cannot imagine a great circle that passes through both of them. (Think of this small circle just below the 'equator' of the sphere and imagine the equator is on a horizontal plane) Could you help me understand this? Or do I misunderstand some concepts here?

[1] Whittlesey, Marshall A.(2020). Spherical Geometry and Its Applications. CRC Press Taylor & Francis Group.

Answer 1

Here's a simple construction, using the fact that three points uniquely determine a plane if not collinear. Take the following three points:

• The given two points.
• The center of the sphere.

The plane determined by these three points should intersect with the sphere with a great circle.

1. The plane should pass through the center of the sphere (so it intersects with the sphere with a great circle) and through the two points provided.
2. The given two points are not antipodal so they are not collinear with the center.