Talk:Henry Ernest Dudeney/Puzzles and Curious Problems/162 - Find the Triangle/Solution

We do have Approximations to Equilateral Triangles by Heronian Triangles, Definition:Fleenor-Heronian Triangle, etc.

Currently Definition:Heronian Triangle coincides with Definition:Rational Triangle, and Equivalence of Definitions of Heronian Triangle with Equivalence of Definitions of Rational Triangle.

However I believe the tricky part of this problem is the possibility that the three side lengths are not consecutive numbers.

(What if we found a triangle with side lengths $11, 13, 14$ and height $12$?)

Those cases could (probably) be resolved using Heron's Formula and bombarding the resulting quartics with Polynomial Factor Theorem.

--RandomUndergrad (talk) 12:18, 11 February 2022 (UTC)