<s>[INST] What is the sum of the maximum and minimum values BC can take? $\small AB$, $\small BD$ and $\small BC$ are are integer measurements
If $\small AB + BD = k$, find the maximum and minimum values that the $\small BC$ side can assume and then add the values found. (Answer: $k$)
My progress:
$\small \triangle ADB: |AB - BD |< AD < AB + BD \therefore \boxed{|AB-BD| < AD < k}\\$ $\small 
\triangle BCD: \boxed{|BD-DC| < BC < BD+DC}\\$ $\small
\boxed{\measuredangle B = 180^o -4\theta}\\$ $\small
\triangle ABC:\dfrac{AB}{\sin\theta}=\dfrac{BC}{\sin3\theta}\rightarrow \boxed{\dfrac{AB}{\sin\theta}=\dfrac{BC}{3\sin\theta-4\sin^3\theta}}\\\small \triangle BCD: \boxed{\frac{BC}{\sin 2\theta}=\frac{BD}{\sin\measuredangle C}}$
These are the relationships I found but I can't "see" how to get the answer.

 [/INST] $AB=a, BD=b, BC=c$
$\triangle ABC: a\sin 3\theta=c\sin \theta \Rightarrow$
$a(3\sin\theta-4\sin^3\theta)=c\sin\theta\Rightarrow$
$a(3-4\sin^2\theta)=c\Rightarrow$
$\sin^2\theta=\frac{3a-c}{4a}\Rightarrow$
$c<3a$
$\triangle BCD: c\sin\angle BCD=b\sin 2\theta \Rightarrow$
$\sin\angle BCD=\frac{b}{c}\sin 2\theta$
$0 < \angle CBD < 180^\circ-4\theta$,
$\angle BCD=180^\circ-\angle CBD-2\theta\Rightarrow$
$2\theta < \angle BCD < 180^\circ-2\theta\Rightarrow$
$\sin\angle BCD > \sin 2\theta\Rightarrow c<b$
Any $c<{\rm min}(3a,b)$ is satisfying conditions.
Addition of positive integer requirements gives
$c_{min}=1$, $c_{max}=k-1$, $c_{min}+c_{max}=k$.
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