2022 IMO Shortlist G5

Let $A B C$ be a triangle, and let $\ell_{1}$ and $\ell_{2}$ be two parallel lines. For $i=1,2$, let $\ell_{i}$ meet the lines $B C, C A$, and $A B$ at $X_{i}, Y_{i}$, and $Z_{i}$, respectively. Suppose that the line through $X_{i}$ perpendicular to $B C$, the line through $Y_{i}$ perpendicular to $C A$, and finally the line through $Z_{i}$ perpendicular to $A B$, determine a non-degenerate triangle $\Delta_{i}$. Show that the circumcircles of $\Delta_{1}$ and $\Delta_{2}$ are tangent to each other. (Vietnam)

设$A B C$ 为三角形，$\ell_{1}$ 和$\ell_{2}$ 为两条平行线。对于$i=1,2$，令$\ell_{i}$分别在$X_{i}、Y_{i}$和$Z_{i}$处与线$B C、C A$和$A B$相交。假设通过$X_{i}$垂直于$B C$的线，通过$Y_{i}$垂直于$C A$的线，最后通过$Z_{i}$垂直于$A B$的线，确定一个非退化三角形$\Delta_{i}$。证明 $\Delta_{1}$ 和 $\Delta_{2}$ 的外接圆彼此相切。 (越南)