A quadratic locus equation in GeoGebra by using the synthetic definition. Dynamic coordinates.

In Step 1 we create three free points A', B' and C'. They should be hidden in the final version of the applet, but for the explanation we just used gray color here.

In Step 2 commands like A=DynamicCoordinates[A',round(x(A')),round(y(A'))] were used to set up constraints for points A, B and C.

In Step 3-4 we define line a and put point D on line a. Then in Steps 5-6 point E is constructed by Euclidean steps (as previously).

Now the locus of point E while point D is moving on line a, is to be defined (Step 7, as previously).

Finally in Step 8 we obtain the equation of the locus.

Now we can also hide points A', B' and C' by right-clicking on them and disable the Show Object option.