Did I discover a function?

Hi, so basically I was messing aroung with the derivatives of x^2 and came across this cool function (well not exatly a function, beause it takes up two different y's at one x) and want to know if it is already known.

So you have the function f(x)=x^(2), the tangent line of which at "a" is

y=f'(a)(x-a)+f(a)=2a(x-a)+a^(2).

If you look for the location "x", where the distance between "x" and "a" is equal to y=2a(x-a)+a^(2), you get the equation

x-a=2a(x-a)+a^(2)

And now if you express "a" in terms of "x" you get:

a1=((1+2x)+(4x^(2)+1)^(1/2))/2 and a2=((1+2x)-(4x^(2)+1)^(1/2))/2

It says that if you have "x", you get the "a", where the "x-a" distance is equal to the "a" locations x^(2) tangent line functions value (y) at a given "x" (so y=2a(x-a)+x^(2))

Finally, if you calculate the distance (x-a, where you substitute "a" with "a1" or "a2"), you get the y=x-a function which looks like this:

purple function, orange is x^2

What I find cool about this function is that it does not depend on "a", so basically it is an endogeneous property of the x^(2) function.

Here is the Desmos file, where Orange is x^2, green is the tangent line, the red vertical line is where "a" is (you have to give it exogeneously, with the slider, and the black vertical line is where "x" is (it is a function of "a"): https://www.desmos.com/calculator/pxq5fijnbn