From

Bezier curve - Wikipedia, the free encyclopedia, the parametric equation for a quadratic Bezier curve is

From

Parabola - Wikipedia, the free encyclopedia, the general equation for a parabola is

and there are further conditions, including

.

I don't know for a fact that a quadratic Bezier curve determines a parabola, but I'll assume it does. Given the condition on

, it cannot assumed that

as you indicate.

Given the 3 points defining the Bezier curve, you want to determine the 6 coefficients of the parabola. Under the invertibility requirement below, you may set the coefficient

arbitrarily to a non-zero number, say

Then you need 5 linear equations to determine the other 5 unknown coefficients.

Let

Then the required 5 linear equations are

To solve these equations requires that the matrix be invertible. That is the invertibility requirement that allows you to set

to an arbitrary non-zero number.

There is a question whether the condition

will be satisfied. I think it must if the Bezier curve does indeed determine a parabola, which as I said I'm assuming is true.

Once you have the coefficients, then which side of the Bezier curve a point

is on is determined by how the sign of

compares to the sign of

where

the middle control point for the Bezier curve.