, 2025

Let $A'B'C'$ denote the reflection of scalene and acute triangle $ABC$ across its Euler-line. Let $P$ be an arbitrary point of the nine-point circle of $ABC$. For every point $X$, let $p(X)$ denote the reflection of $X$ across $P$. a) Let $e_{AB}$ denote the line connecting the orthogonal projection of $A$ to line $BB'$ and the orthogonal projection of $B$ to line $AA'$. Lines $e_{BC}$ and $e_{CA}$ are defined analogously. Prove that these three lines are concurrent (and denote their intersection by $K$). b) Prove that there are two choices of $P$ such that lines $Ap(A')$, $Bp(B')$ and $Cp(C')$ are concurrent, and the four points $p(A)p(A')\cap BC$, $p(B)p(B')\cap CA$, $p(C)p(C')\cap AB$, and $K$ are collinear.