I wrote a computer program that iterates through progressively larger scale triangles. The program seeks whole numbers m & n that represent distances from the base vertices, and when a whole number cevian appears. The cevians are always primes of formula 6n+1 (and multiples of them BUT not including primes not of formula 6n+1).
The first one: cevian=7, triangle side= 8, cevian goes from apex to a point on the edge of the base that is 3 from one base vertex and 5 from the other base vertex.
My results up till cevian length 7^4 are posted in a table here: https://mutaman.neocities.org/
Also recorded are the multiples of rt3 which are orthogonally oriented to these whole number cevians in an adjacent eqilateral triangle (albeit of a different scale). I included them just in case they might be useful in finding a formula to find the next cevian in the sequence.
I am familiar with Stewart's theorem: a^2 = d^2 + m*n where 'a' is the scale, 'd' is the cevian length, and 'm' & 'n' are the distances from either vertex on the base. However I was not able to use it to predict the next cevian, because even though I new the next cevian's value I didn't know the scale of the triangle AND the m or n value. I tried using the values from cevians of smaller scale triangles and used ceva's theorem to combine the cevians but I couldn't find a predictive theory.
Is there a formula to generate these without having to find all cevians brute force and then exclude the ones that aren't whole number?