Incircle, perpendiculars, equality

[ins-]

Let $ABC$ is acute-angled triangle with incircle $k(I)$. Tangency points of $AC$ and $BC$ with $k$ are $X$ and $Y$, respectively. Through $I$ is drawn a line $g$ intersecting $AC$, $BC$ and $k$ at the points $K$, $L$, $M$, $N$ (in this order from left to right). $P$ and $Q$ are the feets of perspendiculars from $X$ and $Y$ to $g$, respectively ($P$ and $Q$ are internal to $k$). Prove that: $\frac{1}{KL} + \frac{1}{QM} = \frac{1}{MN}+\frac{1}{PL}$.