7 Edited tags.
6 added 1052 characters in body

added, Sept 20, 2010. The bounty is for an answer to either question 1 or 2.

I've made partial progress toward 2 using variational methods(direct method of the calculus of variations). I can prove that if a syzygyis chosen anywhere in a neighborhood of binary collision (so $r_{12}(t_c) = \delta$, small, $r_{23} (t_) = r_{13}(t_c) + \delta$)then there exists a brake orbit solutionarc ending in this syzygy and satisfying the inequality of question 2. The proof suggests, but does not prove, that the result holds locally nearisosceles, meaning for brake initial conditionsin a neighborhood of isosceles brake initial conditions ( so
$r_{13} (0) = r_{12} (0) + \epsilon$). If I had uniqueness [modulo rotation and reflection] of brake orbits with specified syzygy endpoints, then my proof would yield a proof of this local version of the alleged theorem.
Unfortunately, my proof does notexclude the possibility of more than one orbit ending in the chosen syzygy, one of which violates the inequality.

5 arxiv tag
4 added 222 characters in body; edited tags; edited tags
3 added tag which seemed appropriate
2 deleted 6 characters in body; edited tags; deleted 6 characters in body
 
1