Earth Precession

From the vector diagram, using bold to denote vectors it follows that

R = r1 + r ,
r = R - r1 , and 
 r = |R - r1| .  Squaring and then taking the square root, 
 r =   Factoring out R.
 r =   Since we are interested in the case in which r > 0, we can invert both sides to get