KöMaL 2026 A936

Given an acute, scalene triangle $ABC$ in the plane. Denote its symmedian point by $K$ and the center of its Feuerbach circle by $N$. Construct, using a compass and straightedge, distinct points $X$, $X^*$, $Y$, $Y^*$ such that $\bullet$ the lines $XY$ and $X^*Y^*$ intersect at $K$; $\bullet$ the lines $XX^*$ and $YY^*$ intersect at $N$; $\bullet$ the points $X$ and $X^*$, as well as $Y$ and $Y^*$, are isogonal conjugates with respect to the triangle.