An interesting analytic geometry problem

[outer hull]

Find the locus of the center of an equilateral triangle inscribed in the parabola y = x^2.

(This problem has bothered me for a long time.
I would appreciate it if somebody could solve it without the assistant of AI)

[languid basin]

Let C(x,y) be the center of the triangle, and A(a,a²) be a point on the parabola. Rotate the vector CA by 120° and by -120°, and impose the condition that the resulting endpoints also lie on the parabola. Solving the resulting system of equations, we obtain the locus ||y=9x²+2||. Btw the radius of the circumcircle centered at (x,y) is 2sqrt(4x²+1).

[outer hull]

thank you

[outer hull]

but how can I simplicity the formula?

there exists a borthering'a'

@languid basin

[outer hull]

i used De Moiver's formula to rotate vectors and then got such ugly thing 😂

[languid basin]

I multiplied the rotation matrix:
1/2 √3/2
-√3/2 1/2
by the vector (a-x, a²-y) and added (x,y) to get the endpoints. And yes, I did some manipulations with polynomials to extract 9x²+2 as a factor. Your (1) and (2) are correct. Just divide (1) by (2) as polynomials in variable a, then factorize the remainder.

[sharp valley]

I ever tried to solve it and no matter what the algebra is horrible, right? I also searched it up in Google and they just mentioned but not show the algebra process.

[languid basin]

Nope, not at all. Well, you do have a bit of brackets expansion and terms manipulations, but nothing really superhuman