Problem (2019KoMaLA729): In a cyclic quadrilateral $ABCD$, the diagonals meet at point $E$, the midpoint of side $AB$ is $F$, and the feet of perpendiculars from $E$ to the lines $DA,AB$ and $BC$ are $P,Q$ and $R$, respectively. Prove that the points $P,Q,R$ and $F$ are concyclic.
AlphaGeometry formalization: a b c = triangle; d = on_circum a b c; e = on_line a c, on_line b d; f = midpoint a b; p = foot e a d; q = foot e a b; r = foot e b c ? cyclic p q r f
HAGeo formalization: A B C = triangle; (ABC) = circle A B C; D = on_circle (ABC); AC = line A C; BD = line B D; E = intersection AC BD; F = midpoint A B; AD = line A D; P = foot E AD; AB = line A B; Q = foot E AB; BC = line B C; R = foot E BC; Prove: concyclic P Q R F
Difficulty score: 2.6