Fregier Quarter Point and the Focus of the Parabola

If you are not familiar with Frégier’s Theorem then you should read my introductory post on the topic “Frégier’s Theorem and Frégier Points”. Later, I also wrote a post about an alternative way of finding the Fregier points for a parabola. The property or theorem discussed in this post is only relevant to parabolas.

I am not sure if the concept of “Fregier Quarter Point” or something similar exists in the current literature. I discovered this special point while playing in GeoGebra. I’ll define the point in the next section.

Fregier Quarter Point Theorem

Definition: Let P be a point on a parabola and let P’ be the corresponding Fregier point. The Fregier Quarter Point ( or Parabola Fregier Quarter Point) is the point P” that is between P and P’ such that PP”/PP’=1/4.

Theorem: The perpendicular to the segment PP’ at the point P” passes through the focus F of the parabola.

Other Properties

in the previous posts I mentioned that the segment PP’ is perpendicular to the parabola tangent at the point P. This means that the line passing trough the Fregier Quarter Point P” and the focus F is parallel to the tangent at point P.

The MathWorld article about the parabola contains the following statement:  the foot of the perpendicular to a tangent to a parabola from the focus always lies on the tangent at the vertex (Honsberger 1995, p. 48).

If we let T be the intersection between the tangent to the parabola at point P and the tangent at the vertex V, then PP”FT is a rectangle.

If we let PR be perpendicular to the tangent to the parabola at the vertex and P’R’ be a similar perpendicular segment (R and R’ being on the vertex tangent) then RR’/TV=4 and RV/TV=2.

4 as a Parabola Constant

The fact that 4 is an important constant related to the parabola is not a surprise. It is well known that the latus rectum of a parabola is 4 times bigger than the distance between the vertex and the focus (4VF). Number 4 also appears in parabola formulas like (y − k)^2 = 4p(x − h). My post about the alternative method to construct the Fregier points also showed some additional properties involving the number 4.

I wonder if somebody can find a connection between the Fregier points and the universal parabolic constant. There are probably many other interesting properties related to the parabola and the other conics that are yet to be discovered.

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