Eight spheres of radius 100 are placed on a flat surface so that each sphere is tangent to two others and their centers are the vertices of a regular octagon. A ninth sphere is placed on the flat surface so that it is tangent to each of the other eight spheres. The radius of this last sphere is \(a +b\sqrt {c},\) where \(a, b,\) and \(c\) are positive integers, and \(c\) is not divisible by the square of any prime. Find \(a + b + c\).

(第十六届AIME1998 第10题)