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Lengths of Sides and Angle Bisectors Are Rational Together

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Let , , be rational numbers and define a triangle by the lengths of its sides:

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,

,

,

which are scaled by their greatest common divisor to be integers.

Let be the incenter of and let , , be the feet of the corresponding Cevians.

Then the lengths of , , are rational.

Conversely, if the lengths of the Cevians are rational, so are the lengths of the triangle.

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Contributed by: Minh Trinh Xuan (August 2022)
Open content licensed under CC BY-NC-SA

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