Bad Karma is a Consequence of Guessing

Since $JI\parallel AB$ , $\;\;HM\parallel CA\;$ and $\;\;KL\parallel CB$ , $\triangle ABC\sim \triangle JIC\sim\triangle MBH\sim \triangle ALK$ $\begin{aligned} \implies \frac{MB}{AB}=\frac{W}{X} ,\;\; \frac{JI}{AB}=\frac{Y}{X} ,\;\; \frac{AL}{AB}=\frac{Z}{X} \;\; \end{aligned}\tag{1}$ Observe, $\triangle ABC\sim\triangle JIC\sim \triangle MFL$ Also, $\triangle JIC$ and $\triangle MFL$ have the same exradius $(X)$ , so we have, $\triangle JIC\cong \triangle MFL\implies JI=LM \tag{2}$ Using $(1)$ and $(2)$ , $\begin{aligned} \frac{MB}{AB}+\frac{JI}{AB}+\frac{AL}{AB}&=\frac{W}{X}+\frac{Y}{X}+\frac{Z}{X} \\ \\ \implies \frac{MB+LM+AL}{AB}&=\frac{W+Y+Z}{X} \\ \\ \therefore \boxed{X=W+Y+Z} \end{aligned}$

The triangles $ABC$ , $ALK$ , $MBH$ and $JIC$ are all similar, and so we deduce that $Z \; = \; \frac{h_a-2X}{h_a}X \hspace{1cm} W \; =\; \frac{h_b - 2X}{h_b}X \hspace{1cm} Y \; = \; \frac{h_c - 2X}{h_c}X$ where $h_a,h_b,h_c$ are the lengths of the altitudes of the triangle $ABC$ from the vertices $A,B,C$ respectively. This is because the length of the altitude from $A$ in the triangle $ALK$ is $h_a - 2X$ , and so on. Thus $\begin{aligned} Y + Z + W & = \; 3X - 2X^2\big(h_a^{-1} + h_b^{-1} + h_c^{-1}\big) \; = \; 3X - \frac{X^2}{\Delta}(a + b + c) \\ & = \; 3X - \frac{2X^2s}{\Delta} \; = \; 3X - 2X \; = \; X \end{aligned}$ where $\Delta,s$ are the area and semiperimeter of the triangle $ABC$ .