No distances, no circles, and you can easily derive an equation.
Just a right angled triangle.
First, the definition of a parabola from the focus and directrix.
Pick a line, the directrix, and a point (B) not on that line (the focus):
Find the line at right angles, passing through a point (C) on that line.
Now find the line from B to C, and the midpoint of BC, which will be D.
Find the line at right angles to BC from D, and the intersection of this line and the vertical line, E, is a point on the parabola.
As point C is moved the parabola is traced out.
The picture is completed with the line BE. Check it!
While looking at the bisection of area formula (see previous posts)
a’b’ = (1/2)ab
where a’ and b’ are the distances from the vertex of points on the sides of the triangle, and a and b are the lengths of the sides I remembered another formula about triangles, the bisection of the angle formula, with a” and b” being the lengths of the two parts into which the opposite side is divided, namely
a”/b” = a/b
These are like Cuba and Puerto Rico, “Two wings of the same bird”, Jose Marti (in Spanish)
Neither involves the angle itself, and so is very general. I decided that there must be a connection, and after a futile look for some duality in the situation I suddenly saw the connection, in simple algebraic terms:
a’b’ = a’/(1/b’)
and so a triangle with sides a’ and 1/b’ will give a”/b” = a’b’
and then a”/b” = (1/2)ab as well.
This is the construction for the halving a triangle.
This is the extended construction for the bisector. The opposite side is in brown.
and this is a gif showing how as the point a’ (E) is moved the brown line (OE, opposite side) moves parallel to itself, thus preserving the value of the ratio a”/b”
And in case you got this far, some light relief. Bullfrog eats dog food, this morning. The dish is 10 inches across.