Hello. Let's suppose that, in the picture, I know how to calculate the measure of the minor arcs AC, CD, AD, being A,C,D elements of S^{2}. I'd like to prove that m(AC)+m(CD) is greater or equal than m(AD), where m(x) is the measure of the minor arc x...I would like to know, how could I start?

I can use Linear Algebra, Modern elemental Geometry, theorems about metric spaces, properties of inverse trigonometric functions, trigonometry, and that kind of things...

I think I could use the usual triangle inequality in the chords of the arcs...but I don't know how to do it.

I forgot, it will be important to say that the three mentioned arcs are in Great circles.