The vertices `A ,Ba n dC` of a variable right triangle lie on a parabola `y^2=4xdot` If the vertex `B` containing the right angle always remains at the point (1, 2), then find the locus of the centroid of triangle `A B Cdot`

Explanation

To find the locus of the centroid of triangle ABC with vertices A, B, and C where B is fixed at (1, 2) and A and C lie on the parabola \(y^2 = 4x\), we can follow these steps:

### Step 1: Identify Points on the Parabola
The parabola \(y^2 = 4x\) can be parametrized using a variable \(T\). Any point on the parabola can be represented as:
\[ A(T_1) = (T_1^2, 2T_1) \quad \text{and} \quad C(T_2) = (T_2^2, 2T_2) \]
where \(T_1\) and \(T_2\) are parameters corresponding to points A and C, respectively.
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