Detect mouse click on a bezier curve's neighborhood

Question

I'm developing a game in HTML5 and JavaScript using Canvas API for drawing graphics. I want to detect if the user has clicked on a bezier curve which has the line width of 20 pixels (something like this one on the picture). I need to validate if the user has clicked on the area drawn by the curve (red and black).

I implemented such an algorithm for lines, but could not adapt it to quadratic and cubic bezier curves. So I need an algorithm that will compare the position of a point relative to a bezier curve which has the form of

B(t) = P₀(1-t)² + P₁(1-t)t + P₂t²

or

B(t) = P₀(1-t)³ + 3*P₁(1-t)t + 3*P₂(1-t)t² + P₃t³

and say if it is in the neighborhood of the bezier curve's path, I mean on the black drawn area.

[Red: Bezier curve's path; Black: Bezier curve's drawb area]

Answer 1

http://stackoverflow.com/questions/2742610/closest-point-on-a-cubic-bezier-curve

In particular, The last answer on the page gives this link, which I thought explained it fairly clearly:

http://jazzros.blogspot.com/2011/03/projecting-point-on-bezier-curve.html?m=1