Integer Triangle of the Gergonne Point

A rational triangle has all three sides rational numbers, and a rational point with respect to a triangle means that the distances from to the vertices of are rational.

If and are rational and , it is possible to construct a rational triangle based on these values such that the Gergonne point is a rational point of [1].

Then we have:

According to the rational isogonal conjugates theorem, since is a rational point of , so is its isogonal conjugate .