Problem (2022TurkeyTSTp3): In a triangle $ABC$, the incircle centered at $I$ is tangent to the sides $BC, AC$ and $AB$ at $D, E$ and $F$, respectively. Let $X, Y$ and $Z$ be the feet of the perpendiculars drawn from $A, B$ and $C$ to a line $\ell$ passing through $I$. Prove that $DX, EY$ and $FZ$ are concurrent.
AlphaGeometry formalization: a b c = triangle; i = incenter a b c; d = foot i b c; e = foot i a c; f = foot i a b; p = free; x = foot a i p; y = foot b i p; z = foot c i p; t = on_line d x, on_line e y ? coll t f z
HAGeo formalization: A B C = triangle; I = incenter A B C; BC = line B C; AC = line A C; AB = line A B; D = foot I BC; E = foot I AC; F = foot I AB; P = point; IP = line I P; X = foot A IP; Y = foot B IP; Z = foot C IP; DX = line D X; EY = line E Y; T = intersection DX EY; Prove: collinear T F Z
Difficulty score: 4.2