Problem (2012ARMOg10p8): The point $E$ is the midpoint of the segment connecting the orthocentre of the scalene triangle $ABC$ and the point $A$. The incircle of triangle $ABC$ incircle is tangent to $AB$ and $AC$ at points $C'$ and $B'$ respectively. Prove that point $F$, the point symmetric to point $E$ with respect to line $B'C'$, lies on the line that passes through both the circumcentre and the incentre of triangle $ABC$.
AlphaGeometry formalization: a b c = triangle; h = orthocenter a b c; e = midpoint a h; i = incenter a b c; b1 = foot i a c; c1 = foot i a b; f = reflect e b1 c1; o = circumcenter a b c ? coll f o i
HAGeo formalization: A B C = triangle; H = orthocenter A B C; E = midpoint H A; I = incenter A B C; AB = line A B; B1 = foot I AB; AC = line A C; C1 = foot I AC; B1C1 = line B1 C1; F = reflect_point_wrt_line E B1C1; O = circumcenter A B C; Prove: collinear O I F
Difficulty score: 3.8