32. In triangle $KLM$, the bisectors $KN$ and $LP$ intersect at point $Q$. Segment $PN$ has a length of 1, and vertex $M$ lies on the circle passing through points $N, P, Q$. Find the sides and angles of triangle $PNQ$.
32. Expressing angle $P Q N$ in terms of the angles of the triangle and considering that $\widehat{P M N}+\widehat{P Q N}=180^{\circ}$, we find $\widehat{P M N}=60^{\circ}$, hence $\widehat{N P Q}=\widehat{Q M N}=$ $=30^{\circ}, \widehat{P N Q}=\widehat{P M Q}=30^{\circ}$, i.e., triangle $P Q N$ is isosceles with angles at side $P N$ each $30^{\circ},|P Q|=|Q N|=\frac{1}{\sqrt{3}}$.