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