Let ABC be a triangle with integral side lengths such that angle A=3 * angle B. Find the minimum value of its perimeter.
(In reply to
re(4): NO DICE.....not a spoiler by xdog)
Bravo
xdog. I think you’ve got it.
I’ve been struggling to find the connection between
the m and n values. But I now realise that,
provided n>m*sqrt(2) (so that a > 0), all integer
pairs (m, n) give valid solutions to my equation (3).
So, writing b = m^{3} (I think you meant m rather than n)
and a + b = mn^{2}, it follows that a = m(n^{2} – m^{2})
and
c = n(n^{2} – 2m^{2}). Also, the perimeter than simplifies to:
a + b + c = n(n – m)(n + 2m)
having its smallest value, 21, when m = 2 and n = 3.
Of course some solutions need rejecting since they
don’t satisfy the triangle inequalities (e.g. m=1, n=2)
Also, some are multiples of others, but such duplicate
similar triangles can be avoided by using coprime
pairs for (m, n).
However, whether this parametric solution covers all
possibilities remains unproven …..

Posted by Harry
on 20150306 19:01:18 