This isn't true, and is obvious when you try to split a bezier without distorting its shape.
As a bezier is just repeated interpolations, the tangents and their lengths can be derived from the intermediate interpolations [1]. This means that if you split a bezier near an existing control point, the new point will have wildly unbalanced tangents on both sides, and yet, connect into the exact same cubic curve.
So both your intuition and your confidence in it are... wrong.
As a bezier is just repeated interpolations, the tangents and their lengths can be derived from the intermediate interpolations [1]. This means that if you split a bezier near an existing control point, the new point will have wildly unbalanced tangents on both sides, and yet, connect into the exact same cubic curve.
So both your intuition and your confidence in it are... wrong.
[1] https://i.stack.imgur.com/I9wKC.png