Reasons for existence of eutectic point for alloy systems
First, there are several other answers that you might be interested in looking at first (full disclosure - they include some by me). 

A question on Physics SE on eutectics

Ideal vs regular solutions and eutectics on Chemistry SE

These together go in to some detail on why the eutectic forms (when the liquid free energy drops down and touches the tie-line between the two solid phases - the Physics one), and the interactions one needs to get the liquid phase to drop below the tie line between the solids (the Chemistry one). To get a eutectic you need an interaction term in the liquid Gibbs free energy.

As for why it happens at such a definite composition, the glib answer is the Gibbs phase rule. In a system at fixed pressure, three phases can coexist at only one point.

Given that this question is, in general, a duplicate of the ones listed above, particularly of the Chemistry SE one, I have no particular need of the bounty. Go read those and see if they help.