unakade

But, how can it be a tetrahedron? The cities are just 5 miles away. Any 4 points on the surface of the earth which are also the vertices of a tetrahedron must be at a distance of 1.63 times the radius of the earth (which is about 6500 miles!!!).

I am not quite sure about this because I did not find numbers a and b which give 216. Also, on wikipedia (http://en.wikipedia.org/wiki/Congruum), a list of first few congrua is given which does not contain 216. So, maybe 216 cannot be represented in this way. I did not find any other flaw in your argument.