Bisected Slopes

Let the two integer slopes be $a$ and $b$ . WLOG let $a>b$ . Let them be bisected by the line having slope $300$ . Now, let the angle between the former two lines be $\theta$ . Using standard formulae, we have

$\tan\frac{\theta}{2}=\frac{a-300}{1+300a}=\frac{300-b}{1+300b}$

Hence, we get after some manipulations, $600ab-89999\left(a+b\right)=600$

To solve this, multiply $600$ to both sides of the equation and let $600a=x$ and $600b=y$ .

So we simply need to find the integral solutions of the equation
$\left(x-89999\right)\left(y-89999\right)=8100180001$

We can simply start checking all possible integer solutions, but the first attempt will give the answer. Taking $x=90000$ and $y=8100270000$ we get the answer $\displaystyle \left(a,b\right)\ =\ \left(150,13500450\right)$

Let the missing slopes be $m_1$ and $m_2$ . Then

$m_1=\dfrac {89999m_2+600}{600m_2-89999}$

For integral value of $m_1$ , $m_2=150$ and then $m_1=13500450$ , so that the sum of the slopes is $\boxed {13500600}$ .