Hi,
Simplistically, EIGRP will take all of the route advertisements received from its neighbors and pick the one with the lowest metric and install it as the best route. The metric advertised for the route by each of the neighbors is the advertised distance associated with that neighbor. The FD for the route will be the lowest AD + the metrics associated with the incoming interface. This is not a straight summation - the bandwidth component of the metric is set to be:
mim(bandwidth metric received from neighbor, bandwidth configured on interface over which the route was received).
The delay component of the metric is set to be:
(delay metric received from neighbor) + (delay configured on interface over which the route was received)
Once it has done that, it will look at all the other metrics advertised by the other neighbors. If the AD for that route received from a particular neighbor is less than the FD calculated by this router, that neighbor becomes a feasible successor for that route.
Now to your question on how EIGRP decides that the route is loop-free. For a route to be considered loop-free, the AD received from that neighbor has to be lower than the FD calculated by this router. If the AD reported by the neighbor is lower than the FD (which is the lowest metric that the calculating router knows), that neighbor will never use this (i.e. the calculating router) as the next-hop for that router - why would a neighbor use a higher-cost route than a better one that it already has. Therefore, this condition that the AD be less than the FD ensures the loop-free property.
Hope that helps - pls rate posts that help.
Paresh.
