Lengths of Sides and Angle Bisectors Are Rational Together

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