The Perfect Triangle

In Chapter D problem 21 of Richard Guy's wonderful book called "Unsolved Problems in Number Theory" he asks the following:

Does there exist a triangle with integer sides, medians and area?

Why do we call it perfect? It turns out that the description above forces a perfect triangle to also have 3 rational altitudes, 3 rational angle bisectors, 3 rational orthogonal side dividers, as well as a host of other rational lengths.

One the one hand, no-one has ever found a perfect triangle---despite lots of computational searching (see this paper for more details). On the other hand, no-one has ever proven that a perfect triangle cannot exist. Some have tried, for example see page 20-22 of Schubert, but the proofs are all incorrect.