Problem (2022G5): Let $ABC$ be a triangle and $\ell_1,\ell_2$ be two parallel lines. Let $\ell_i$ intersects line $BC,CA,AB$ at $X_i,Y_i,Z_i$, respectively. Let $\Delta_i$ be the triangle formed by the line passed through $X_i$ and perpendicular to $BC$, the line passed through $Y_i$ and perpendicular to $CA$, and the line passed through $Z_i$ and perpendicular to $AB$. Prove that the circumcircles of $\Delta_1$ and $\Delta_2$ are tangent.
AlphaGeometry formalization: a b c = triangle; x1 = on_line b c; y1 = on_line a c; z1 = on_line a b, on_line x1 y1; x2 = on_line b c; y2 = on_line a c, on_pline x2 x1 y1; z2 = on_line a b, on_pline x2 x1 y1; u1 = on_tline y1 a c, on_tline z1 a b; v1 = on_tline x1 b c, on_tline z1 a b; w1 = on_tline x1 b c, on_tline y1 a c; u2 = on_tline y2 a c, on_tline z2 a b; v2 = on_tline x2 b c, on_tline z2 a b; w2 = on_tline x2 b c, on_tline y2 a c; o1 = circumcenter u1 v1 w1; o2 = circumcenter u2 v2 w2; t = on_circle o1 u1, on_circle o2 u2 ? coll t o1 o2
HAGeo formalization: A B C = triangle; BC = line B C; X1 = on_line BC; AC = line A C; Y1 = on_line AC; l1 = line X1 Y1; AB = line A B; Z1 = intersection l1 AB; X2 = on_line BC; l2 = parallel_line X2 l1; Y2 = intersection l2 AC; Z2 = intersection l2 AB; l3 = perpendicular_line X1 BC; l4 = perpendicular_line Y1 AC; l5 = perpendicular_line Z1 AB; U1 = intersection l4 l5; V1 = intersection l3 l5; W1 = intersection l3 l4; l6 = perpendicular_line X2 BC; l7 = perpendicular_line Y2 AC; l8 = perpendicular_line Z2 AB; U2 = intersection l7 l8; V2 = intersection l6 l8; W2 = intersection l6 l7; O1 = circumcenter U1 V1 W1; O2 = circumcenter U2 V2 W2; (O1) = circle_center_point O1 U1; (O2) = circle_center_point O2 U2; T = intersection (O1) (O2); Prove: collinear T O1 O2
Difficulty score: 2.9