# Curved lines on a paper [closed]

There are curved lines drawn on a piece of paper. Lines can have two characteristics:

1. One of its endpoints is on another characteristic 1 line
2. One of its endpoints is on another characteristic 2 line, exactly halfway between:
• Its two endpoints
• A point where a characteristic 2 line touches it and an endpoint
• Two points where a characteristic 2 line touches it

Is it possible for there to be a third characteristic such that, for all sets of lines

• It is impossible for all possible characteristic 3 starting points connect to a unique characteristic 1 endpoint
• It is impossible for all possible characteristic 2 starting points connect to a unique characteristic 3 starting point
• I'm having a hard time understanding this question. You say that lines can have characteristic 1 or characteristic 2. What is a "characteristic 2 starting point"? Is "characteristic 3" something that lines can have? Can a line have both characteristics 1 and 2? What about 1 and 3, or 2 and 3? – f'' Sep 25 '15 at 0:53
• It may be helpful to replace the "characteristics" with colors (assuming each line gets one characteristic) and drawing an example picture. – Deusovi Sep 25 '15 at 0:58