I know this is a popular problem for algebra; I am looking for a constructive way to find the normal(s) to a parabola across an external point.
The obvious guess is that one needs to find the circle that contains all the foots. I have found in the literature a constructive recipe but it uses algebra, basically one takes coordinates centered on the vertex, horizontal abscisa being the axis of the parabola, and then to find the circle:
- The vertical cordinate is 1/4 of the coordinate of the external point A
- The horizontal coordinate is the focal distance OF plus half the coordinate of A.
Well, it works, but it is sort of unsatisfactory, as it clearly uses the knowledge of the algebraic solution without any intuition from the properties of conics. I wonder which other approaches exist.