Question 127
On the sides $AB$, $BC$ and $CA$ of triangle $ABC$, points $L$, $M$ and $N$ are chosen, respectively, such that the lines $CL$, $AM$ and $BN$ intersect at a common point $O$ inside the triangle and the quadrilaterals $ALON$, $BMOL$ and $CNOM$ have incircles. Prove that
$$
\frac{1}{AL \cdot BM} + \frac{1}{BM \cdot CN} + \frac{1}{CN \cdot AL} = \frac{1}{AN \cdot BL} + \frac{1}{BL \cdot CM} + \frac{1}{CM \cdot AN}.
$$
$$
\frac{1}{AL \cdot BM} + \frac{1}{BM \cdot CN} + \frac{1}{CN \cdot AL} = \frac{1}{AN \cdot BL} + \frac{1}{BL \cdot CM} + \frac{1}{CM \cdot AN}.
$$