Step 1: Derive it:
Use $\tan\frac A2=\sqrt{\frac{(s-b)(s-c)}{s(s-a)}}$ and $\tan\frac B2=\sqrt{\frac{(s-c)(s-a)}{s(s-b)}}$. Their product is $\sqrt{\frac{(s-c)^2}{s^2}}=\frac{s-c}{s}$.
Step 2: Test with a triangle:
Take an equilateral triangle with $a=b=c=2$. Then $\tan30^\circ\tan30^\circ=\tfrac13$, so the left side is $\tfrac43$. And $s=3$.
Step 3: Find k:
$k=\tfrac43\times3=4=a+b$. Check others: $2\ne4$, $a+b-c=2\ne 4$, $s-c=1\ne4$.
Final Answer:
An equilateral test triangle gives k = 4 = a + b.
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