In chapter D problem 21 of Richard Guy's wonderful book called "Unsolved Problems in Number Theory" he writes the following:
Definition : A "perfect triangle" is a triangle with integer
sides, medians and area.
Why is it called perfect? It turns out that the definition 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 lengths.
One the one hand, noone has ever found a perfect triangledespite
lots of computational searching (see this paper
for more details).
On the other hand, noone has ever proven that a perfect triangle cannot exist.
Some have tried, for example see page 2022 of
Schubert, but the proofs are all incorrect.
