Problem (2019KoMaLA743): The incircle of tangential quadrilateral $ABCD$ intersects diagonal $BD$ at $P$ and $Q$ $(BP<BQ).$ Let $UV$ be the diameter of the incircle perpendicular to $AC$ $(BU<BV).$ Show that the lines $AC,PV,$ and $QU$ pass through one point.
AlphaGeometry formalization: t1 t2 t3 = triangle; i = circumcenter t1 t2 t3; t4 = on_circle i t1; a = on_tline t1 i t1, on_tline t2 i t2; b = on_tline t2 i t2, on_tline t3 i t3; c = on_tline t3 i t3, on_tline t4 i t4; d = on_tline t1 i t1, on_tline t4 i t4; p = on_line b d, on_circle i t1; q = on_line b d, on_circle i t1; u = on_tline i a c, on_circle i t1; v = on_tline i a c, on_circle i t1; x = on_line p u, on_line q v ? coll a c x
HAGeo formalization: T1 T2 T3 = triangle; I = circumcenter T1 T2 T3; (I) = circle_center_point I T1; T4 = on_circle (I); IT1 = line I T1; IT2 = line I T2; IT3 = line I T3; IT4 = line I T4; l1 = perpendicular_line T1 IT1; l2 = perpendicular_line T2 IT2; l3 = perpendicular_line T3 IT3; l4 = perpendicular_line T4 IT4; A = intersection l1 l2; B = intersection l2 l3; C = intersection l3 l4; D = intersection l4 l1; BD = line B D; P Q = intersection BD (I); AC = line A C; γ1 = perpendicular_line I AC; U V = intersection (I) γ1; PU = line P U; QV = line Q V; X = intersection PU QV; Prove: collinear A C X
Difficulty score: 2.8