Bezier curve conundrums

I have a question relating to computation of bezier curves - rather showing my ignorance of maths I'm afraid...

I'm working in 2D with points and bezier curves.

I want the user to be able to insert a new point into an existing bezier curve - thus creating two curves which accurately mirror the shape of the original. How the devil can I calculate this? Luckily, I'm only using 3point beziers - but might have to move to 4 (max)!

Any ideas?

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