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$\small AB$, $\small BD$ 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.

enter image description here

peta arantes
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    Not sure I understand. Are $A$ and $D$ fixed points? – Brian Tung Jul 19 '21 at 18:31
  • @BrianTung +1, I also am confused. To the OP: +1 to your question for work shown, embedding a clarifying diagram into the question, and for making a serious effort to explain the problem. However, I agree with Brian Tung: it is unclear what you are asking. Please proofread your question and try to edit it so that it is (ideally) absolutely impossible for anyone to be in any way confused about what you are asking. Often, with a complicated question, such editing can be challenging. – user2661923 Jul 19 '21 at 18:42
  • Yes..A and D are fiexd points.. I have two triangles..ABC e BCD, AB + BD = k.. I need find maiximum e minimum value that BC can assume. and then add the two values. The triangle sides are integer values – peta arantes Jul 19 '21 at 19:38
  • @useer2661923 Sorry, I tried to explain it better now.. – peta arantes Jul 19 '21 at 19:50
  • There are two larger and three smaller triangles. Do you mean all sides of each are integers, or just the larger ones? Please specify. – coffeemath Jul 19 '21 at 20:03
  • @user2661923 The problem only mentions AB, BD and BC with integers. It's a peruvian problem. – peta arantes Jul 19 '21 at 21:53
  • B is on an ellipse with A and D as ellipse foci. – Moti Jul 20 '21 at 05:54
  • @user2661923 This problem follows the same idea, maybe help: https://math.stackexchange.com/questions/4200250/find-the-largest-and-smallest-values-of-the-angles-of-a-triangle?noredirect=1#comment8714595_4200250 – peta arantes Jul 20 '21 at 13:31
  • @BrianTung This problem follows the same idea, maybe help: https://math.stackexchange.com/questions/4200250/find-the-largest-and-smallest-values-of-the-angles-of-a-triangle?noredirect=1#comment8714595_4200250 – peta arantes Jul 20 '21 at 13:32
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    @coffeemath The problem only mentions AB, BD and BC with integers. It's a peruvian problem. – peta arantes Jul 20 '21 at 13:33
  • Why the use of "sen" - do you mean the simple sine function? Your equation in the Rectangle does not seem right. – Moti Jul 20 '21 at 16:37
  • @Moti ..Why is it not correct? È the simple application of the sine theorem... – peta arantes Jul 20 '21 at 16:57
  • Because both BC and BD are above the line - the sine theorem – Moti Jul 20 '21 at 17:00
  • Let us continue this discussion in chat. – peta arantes Jul 20 '21 at 18:07
  • I see you corrected it. – Moti Jul 21 '21 at 06:43

1 Answers1

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$AB=a, BD=b, BC=c$

$\triangle ABC: a\sin 3\theta=c\sin \theta \Rightarrow$ $a(3\sin\theta-4\sin^3\theta)=c\sin\theta\Rightarrow$ $a(3-4\sin^2\theta)=c\Rightarrow$
$\sin^2\theta=\frac{3a-c}{4a}\Rightarrow$ $c<3a$

$\triangle BCD: c\sin\angle BCD=b\sin 2\theta \Rightarrow$ $\sin\angle BCD=\frac{b}{c}\sin 2\theta$

$0 < \angle CBD < 180^\circ-4\theta$, $\angle BCD=180^\circ-\angle CBD-2\theta\Rightarrow$ $2\theta < \angle BCD < 180^\circ-2\theta\Rightarrow$ $\sin\angle BCD > \sin 2\theta\Rightarrow c<b$

Any $c<{\rm min}(3a,b)$ is satisfying conditions. Addition of positive integer requirements gives $c_{min}=1$, $c_{max}=k-1$, $c_{min}+c_{max}=k$.

Ivan Kaznacheyeu
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