Problem (2024ShuZhiMiMOM0p3): Given rhombus $ABCD$ and point $E$, let $EB,EC$ meet $DA$ at $F,G$. The common external tangents (other than $DA$) of incircles of $\triangle ABF$ and $\triangle CDG$ meet $AB,CD$ at $H,K$ and $EB,EC$ at $I,J$. Prove the Newton lines of quadrilaterals $AHIF$, $DKJG$, $BCJI$ are concurrent.
AlphaGeometry formalization: b a c = iso_triangle; d = parallelogram a b c d; f = on_line d a; g = on_line d a; i1 = incenter a b f; i2 = incenter c d g; t1 = foot i1 a d; t2 = foot i2 a d; t3 = reflect t1 i1 i2; t4 = reflect t2 i1 i2; h = on_line t3 t4, on_line a b; i = on_line t3 t4, on_line b f; j = on_line t3 t4, on_line c g; k = on_line t3 t4, on_line c d; m1 = midpoint a i; m2 = midpoint h f; m3 = midpoint d j; m4 = midpoint k g; m5 = midpoint b j; m6 = midpoint c i; p = on_line m1 m2, on_line m3 m4 ? coll p m5 m6
HAGeo formalization: B A C = isos_triangle; D = parallelogram A B C; AD = line A D; F = on_line AD; G = on_line AD; I1 = incenter A B F; I2 = incenter C D G; T1 = foot I1 AD; T2 = foot I2 AD; I1I2 = line I1 I2; T3 = reflect_point_wrt_line T1 I1I2; T4 = reflect_point_wrt_line T2 I1I2; T3T4 = line T3 T4; AB = line A B; H = intersection T3T4 AB; BF = line B F; I = intersection T3T4 BF; CG = line C G; J = intersection T3T4 CG; CD = line C D; K = intersection T3T4 CD; M1 = midpoint A I; M2 = midpoint H F; M3 = midpoint D J; M4 = midpoint K G; M5 = midpoint B J; M6 = midpoint C I; M1M2 = line M1 M2; M3M4 = line M3 M4; P = intersection M1M2 M3M4; Prove: collinear P M5 M6
Difficulty score: 6.3