A perfect triangle is defined by Richard K. Guy in problem D21 in "Unsolved Problems in Number Theory" as a triangle with integer sides, integer medians and integer area. One would think that given all these constraints that by now someone would have either found such a triangle or proven that they cannot exist. However this is not the case. The main reason is our fundamental lack of understanding of the nature of rational points on elliptic surfaces one of which is shown above. In fact, any rational point on the above surface would correspond to a triangle with three rational medians. If it also has rational area then it would be perfect.

For more details see Triangles with 3 rational medians.