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Minimizing Perimeter (Posted on 2015-03-04) Difficulty: 2 of 5
Let ABC be a triangle with integral side lengths such that angle A=3 * angle B. Find the minimum value of its perimeter.

No Solution Yet Submitted by Danish Ahmed Khan    
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Some Thoughts re(5): NO DICE.....not a spoiler | Comment 14 of 15 |
(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 = m3 (I think you meant m rather than n)

and a + b = mn2, it follows that a = m(n2 – m2) and

c = n(n2 – 2m2). 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 co-prime
pairs for (m, n).

However, whether this parametric solution covers all
possibilities remains unproven …..

  Posted by Harry on 2015-03-06 19:01:18

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