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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(3): NO DICE.....not a spoiler | Comment 12 of 15 |
(In reply to re(2): NO DICE.....not a spoiler by Ady TZIDON)

Yes, I’m also keen to find an analytical method Ady,
but, because the sides need to be integers, we’re facing
a Diophantine problem and not one with a continuous
variable that can be solved with calculus. Also, since
the perimeter can be integral without the sides being
integral, a, b and c must feature in the analysis.
The best I can do is find the Diophantine equation, but
sadly not solve it.
Sine rules give:
   a/b = sin(3B)/sin(B)  and c/b = sin(4B)/sin(B)
which can be reduced, respectively, to

    a/b = 4*cos2(B) – 1 = 1 + 2*cos(2B)               (1)

    c/b = 4*cos(B)cos(2B)                                    (2)

Squaring (2) and using (1) to substitute for cos(B)

and cos(2B):    c2/b2 = (a/b + 1)(a/b – 1)2

Thus:                bc2 = (a + b)(a – b)2                   (3)

A computer search shows that if we allow only solutions
that obey the triangle inequalities viz. a + b > c,
b + c > a, c + a > b, then  we are left with exactly
those that Charlie found earlier, so we are on the right
It’s not difficult to prove from (3) that if we allow only
primitive solutions (with GCD(a, b, c) = 1) then a and b
will have a common factor and b | a^3 (perhaps b itself
is always a cube??) so the computer search can be
streamlined, but I still can’t find a way of opening it up
analytically. Help needed!

  Posted by Harry on 2015-03-05 15:53:07

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