Lines Through a Triangle's Points

Difficulty: Medium · Topics: Discrete Mathematics, Combinations, Geometry

Nine points A, B, C, …, K are marked on the sides of a triangle, as shown.

graph "x":[-0.8,4.8],"y":[-0.7,4.2],"axes":false,"ink":true,"label":"Triangle with vertices A, E, H; B, C, D on side AE; F, G on side EH; K on side AH","items":["type":"polygon","points":[[2.0,3.6],[0.0,0.0],[4.0,0.0]],"type":"point","at":[2.0,3.6],"label":"A","anchor":"n","type":"point","at":[0.0,0.0],"label":"E","anchor":"sw","type":"point","at":[4.0,0.0],"label":"H","anchor":"se","type":"point","at":[1.5,2.7],"label":"B","anchor":"w","type":"point","at":[1.0,1.8],"label":"C","anchor":"w","type":"point","at":[0.5,0.9],"label":"D","anchor":"w","type":"point","at":[1.32,0.0],"label":"F","anchor":"s","type":"point","at":[2.68,0.0],"label":"G","anchor":"s","type":"point","at":[2.8,2.16],"label":"K","anchor":"e"]

At most how many different lines can be drawn through these points?