Problem

Source: 2018 Korea Winter Program Practice Test 1 #3

Tags: Simson line, Concyclic, geometry



Denote $A_{DE}$ by the foot of perpendicular line from $A$ to line $DE$. Given concyclic points $A,B,C,D,E,F$, show that the three points determined by the lines $A_{FD}A_{DE}$ , $B_{DE}B_{EF}$ , $C_{EF}C_{FD}$, and the three points determined by the lines $D_{CA}D_{AB}$ , $E_{AB}E_{BC}$ , $F_{BC}F_{CA}$ are concyclic.