KöMaL 2026 A925

Call four points to be in general position if they are pairwise distinct, no three of them are collinear, and no two of the six lines determined by them are parallel. Let $A$, $B$, $C$, $D$ be points in general position lying on a circle. Let $E$ be the intersection point of lines $AB$ and $CD$, let $F$ be the intersection point of lines $AC$ and $BD$, and let $G$ be the intersection point of lines $AD$ and $BC$. In triangle $EFG$, denote by $P$, $Q$, and $R$ the feet of the altitudes corresponding to the vertices $E$, $F$, and $G$, respectively. a) Show that the triples of lines $AR$-$BQ$-$CP$, $AQ$-$BR$-$DP$, $AP$-$CR$-$DQ$, and $BP$-$CQ$-$DR$ are either concurrent or parallel. b) Suppose that each of the four triples of lines is concurrent, and denote their points of concurrency by $X$, $Y$, $Z$, and $W$, respectively. Assume further that these points are in general position. Prove that the Miquel points of quadrilaterals $XWYZ$, $XYZW$, and $XZWY$ are precisely the points $P$, $Q$, and $R$.