Check out linear referencing... What is linear referencing?—Help | ArcGIS Desktop
Your line feature(s) will need M-values and a unique id to create routes. You should then be able to use the Make Route Event Layer tool to generate points off your table.
Densify—Help | ArcGIS Desktop only allows for equal incrementing.
How many points do you have to add? 10? 100? (ie, manual versus scripting.
Is your line straight? (image would help)
How is your python?
You need to provide some use-case data or the problem is too simple
dist = [1.6, 3.9, 10.5, 12.7] hence x = [1.6, 3.9, 10.5, 12.7] y = [0.0, 0.0, 0.0, 0.0] Rotate the segment