VTMO 2026 - Grade 9 - Final - SET 4 - Self-study Question 2

Let $(I)$ be the incircle of a triangle $(S)$. Three tangents to $(I)$, parallel to the three sides of $(S)$, meet the sides of the triangle at six points $A,B,C,D,E,F$. Prove that opposite sides of hexagon $ABCDEF$ have equal lengths.