A very rational tetrahedron

[Ed Pegg]

Here's a nice tetrahedron 
{{0, 0, 0}, {2, 0, 0}, {0, 4, 0}, {0, 3, 4}}    

A lot of usually nasty centers are rational
{{"Incenter", {8/13, 16/13, 8/13}}, {"Centroid", {1/2, 7/4, 1}},
{"Circumcenter", {1, 2, 13/8}},
{"Monge", {0, 3/2, 3/8}}, {"12Point", {1/3, 5/3, 19/24}},
{"Symmedian", {64/93, 74/93, 32/93}}, {"Spieker", {6/13, 25/13, 44/39}}}  

The incenter is based on face areas, so unless all the faces have integer area, the incenter will be highly complicated.  

For the vertices to be integers, all the faces need to be heronian, but edges can be square roots.  Doing that with different scalene triangles is non-trivial... and then the Gram determinant has to be a perfect square The only set of edges that worked for max edge<sqrt(30)  was  {4,16,25,20,29,17}  
and {{0, 0, 0}, {2, 0, 0}, {0, 4, 0}, {0, 3, 4}}   is a representation.    

I'm not sure yet if there's a non-messy way to find more of these. There's 
probably a sequence driven by these.   {{0, 0, 0}, {2, 0, 0}, {0, 4, 0}, {0, 3, 4}}  
looks very simple but was difficult for me to find.