The is year's IMO problem 6 was a geometry problem that only 6 participants managed to solve completely. The problem is formulated like this:
Let $ABC$ be an acute triangle with circumcircle $\Gamma$. Let $\ell$ be a tangent line to $\Gamma$, and let $\ell_a$, $\ell_b$ and $\ell_c$ be the lines obtained by reflecting $\ell$ in the lines $BC$, $CA$ and $AB$, respectively
Show that the circumcircle of the triangle determined by the lines $\ell_a$, $\ell_b$ and $\ell_c$ is tangent to the circle $\Gamma$.
A few solutions were found to this problem: using inversions, complex numbers, angle chasing, etc. My question is if we can reduce the problem to a simpler one in the following way:
Can we construct a triangle $\Delta$ for which $\Gamma$ is the incircle and $\Gamma_1$ is the 9 point circle? Of course, the answer should be yes if the circles are tangent and the radius of $\Gamma_1$ is greater than the radius of $\Gamma$. In this way we just apply a well known theorem of Feuerbach which says that the incircle and 9 point circle are tangent. How could we construct the triangle $\Delta$, starting from $ABC$?
This was my first idea when I saw the problem but didn't manage to finalize it.