<s>[INST] Elegant way of finding the least perimeter of triangle A triangle $ABC$ has positive integer sides, $\angle A = 2\angle B$ and $\angle C > \pi/2$ , then the minimum length of the perimeter of $ABC$ is?
We have $\angle A = 2\angle B$
$\Rightarrow \sin A=\sin 2B=2 \sin B \cos B $
$\sin C=\sin(\pi-3B)=\sin(3B)=3\sin B-4\sin^3B$
Using $\frac{\sin A}{a}=\frac{\sin B}{b}=\frac{\sin C}{c}$
$a=2b\cos B $
$c=b(3-4\sin^2 B)$
From above two equations $a^2=b(c+b) $
NOTE: similar questions have been asked earlier but I want to do this question analysing with $a^2=b(c+b)$ and no fancy inequalities
 [/INST] First of all, it is consider very bad style to just throw random equations around.  Write down in English what exactly you are doing --- is it an assumption you make?  A given condition?  Some logical deduction from earlier? And every sentence should start with an English word not an equation, unless you absolutely have to.
Note that it suffices to consider $a,b,c$ coprime (otherwise you get a smaller triangle by scaling).  Now $a^2=b^2+bc$ is equivalent to $c^2+4a^2=(2b+c)^2$.  If $c$ is even we have $a^2+(\frac12c)^2=(b+\frac12c)^2$, and we must have $a$ odd (otherwise $a,c$ even gives $b$ also even, and so not primitive).  So applying the classification of primitive Pythagorean triples, we have:
\begin{align*}
(c,2a,2b+c)&=(m^2-n^2,2mn,m^2+n^2)\text{ or }\\
(\tfrac12c,a,b+\tfrac12c)&=(2mn,m^2-n^2,m^2+n^2)\\
\end{align*}
for some $m,n$ coprime, $m>n$ opposite parity.  Now analyse each case separately:
Case 1: $(c,2a,2b+c)=(m^2-n^2,2mn,m^2+n^2)$, so $(a,b,c)=(mn,n^2,m^2-n^2)$.  We want $c^2>a^2+b^2$, so $m^2>3n^2$.  Also triangle inequality $c<a+b$ gives $(m+n)(m-2n)<0$.  So seek $\sqrt3<\frac{m}n<2$ (it is probably not a good idea to bring in $\sqrt3$ but you know what I mean) and we will have perimeter $m(m+n)$.  $\frac{m}n=\frac74$ is obviously the best candidate here with least $n$ and least $m$.  So we have $(a,b,c)=(28,16,33)$ and perimeter $77$.
Case 2: $(\tfrac12c,a,b+\tfrac12c)=(2mn,m^2-n^2,m^2+n^2)$ so $(a,b,c)=(m^2-n^2,(m-n)^2,4mn)$.  Then $c^2>a^2+b^2$ gives $16m^2n^2>(m+n)^2(m-n)^2+(m-n)^4$, i.e., $(m+n)^2 (m^2-4mn+n^2)<0$ and $c<a+b$ gives $m>3n$, so $3<\frac{m}{n}<2+\sqrt3$ and the perimeter is $2m(m+n)$.  The best candidate here is the choice $\frac{m}n=\frac72$, which gives $(a,b,c)=(45,25,56)$ and perimeter $126$.
So the least perimeter is 77.
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