N is the north pole. A and B are points on a great circle through N equidistant from N. C is a point on the equator. Show that the great circle through C and N bisects the angle ACB in the spherical triangle ABC (a spherical triangle has great circle arcs as sides).
Solution
Let SA, SB, SN be the great circles through A and C, B and C, and N and C respectively. Let C' be the point directly opposite C on the sphere. Then any great circle through C also goes through C'. So, in particular, SA, SB and SN go through C'.
Two great circles through C meet at the same angle at C and at C', so the spherical angles ACN and AC'N are equal. Now rotate the sphere through an angle 180o about the diameter through N. Then great circles through N map into themselves, so C and C' change places (C is on the equator). Also A and B change places (they are equidistant from N). SA must go into another great circle through C and C'. But since A maps to B, it must be SB. Hence the spherical angle AC'N = angle BCN (since one rotates into the other). Hence ACN and BCN are equal.
© John Scholes
jscholes@kalva.demon.co.uk
20 Aug 2002