[Math] Why is there always a great circle through two points on the sphere

Question

I have the following understanding of this: the plane that contains the two points and the centre of the sphere cuts the sphere into two equal parts. The intersection of the plane and the sphere is the great circle we are looking for.

But I am failing to show why this works.An intuitive proof that I can present to an audience who has no knowledge of calculus would be helpful.

Answer 1

Note that there is a plane (and it is unique if the two points are not antipodal) which passes through the two given distinct points and the center of the sphere. The intersection of this plane and the sphere gives the great circle (each plane cut a the sphere along a circle, if it passes through the center then its radius is the radius of the sphere).