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Integer Triangle of the Gergonne Point

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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.

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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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Contributed by: Minh Trinh Xuan (June 13)
Open content licensed under CC BY-NC-SA

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Reference

[1] C. Kimberling. "Encyclopedia of Triangle Centers." (Dec 29, 2022) faculty.evansville.edu/ck6/encyclopedia.

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