What is minimum value of $l$ where $l$ is one of the prependiculer line of triangle

Question

$l,k,2$ are length of prependicular line of triangle$ABC$.Where $\frac{k+2}{k-2}$ is not integer ,but rational number, Integer $l$'s maximum value is $f(k)$, find $f(k)$

I didn't have an idea to solve this problem. Please give some hint...

Answer 1

First, we set the Area of the triangle be “lkt”. Then the three sides of the triangle are 2kt, 2lt, and lkt. So, we have 2k+2l>lk, 2k+lk>2l, and 2l+lk>2k. To these, we have 2k>l(k-2), 2k>l(2-k), and (2+k)l>2k. if k is more than 2, l is less than 2k/(k-2). If k is less than 2, l is less than 2k/(2-k). if k is 2, l just needs to be more than 1.