Finding the Equations of Angle Bisectors The question is as follows: 

Let $K(5, 12)$, $L(14, 0)$ and  $M (0,0)$. The line $x + 2y = 14$ bisects angle $MLK$. Find equations for the bisectors of the angles $KML$ and $MKL$. 

Any help will be truly appreciated! 

Let $KD$ be bisector of $\Delta KLM$.
Thus, since $$\frac{LD}{DM}=\frac{KL}{KM}=\frac{\sqrt{9^2+12^2}}{\sqrt{5^2+12^2}}=\frac{15}{13},$$
we obtain
$$D\left(\frac{13\cdot14+0}{15+13},0\right)$$ or
$$D(6.5,0).$$
Thus, $$m_{KD}=\frac{12-0}{5-6.5}=-8$$ and for the equation of $KD$ we obtain:
$$y-12=-8(x-5)$$ or
$$y=-8x+52.$$
Now, $y=-8x+52$ and $x+2y=14$ intersect in the point $I(6,4)$.
Thus, for the equation of the third bisector we obtain:
$$y-0=\frac{4-0}{6-0}(x-0)$$ or
$$y=\frac{2}{3}x.$$
Done!
