The 3 black points are where they live, and the red points are the 
possible places for the banquet.

It looks as though the red point MC(6,8) Morgan City is the one that's
equidistant from the three black points:



Now we will find algebraically a point which is equidistant from
the three points where they live, WITHOUT ASSUMING that it is
necessarily at Morgan City MC(6,8).

Let the point P(p,q) be a point which is equidistance from all three
black points.  Then using the distance formula



Distance from BS(1,3) to P(p,q)



Distance from RF(11,3) to P(p,q)



Distance from BM(7,15) to P(p,q)



Setting the first two distances equal:



Square both sides:

















Setting the first and third distances equal:



Substituting 



Squaring both sides:

















So the point which is equidistant from the three towns
where the players live is P(p,q) or P(6,8), and we see that
this really is the point at Morgan City MC(6,8).

Edwin