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.