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 .


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[1] C. Kimberling. "Encyclopedia of Triangle Centers." (Dec 29, 2022)
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