Question 175
Circles $w_1$ and $w_2$ intersect at the points $P$ and $Q$. Suppose $AB$ and $CD$ are parallel diameters of the circles $w_1$ and $w_2$ respectively. Moreover, none of the points $A, B, C, D$ coincides with $P$ or $Q$, and the points are located on the circles in the following order: $A, B, P, Q$ on $w_1$ and $C, D, P, Q$ on $w_2$. Lines $AP$ and $BQ$ intersect at point $X$ and lines $CP$ and $DQ$ intersect at $Y \neq X$. Prove that, regardless of the choice of parallel diameters $AB$ and $CD$, all the lines $XY$ intersect at one point or are parallel.
(Nazar Serdiuk)
Figure 26
(Nazar Serdiuk)
Figure 26