QA-2Co1: Pair of Circumscribed QA-Parabolas

Two parabolas can be constructed from 4 points.

There are however some limitations to the positioning of these points for having a real solution of these parabolas.

1. No 3 of the 4 points must be collinear.
2. A Quadrangle consists of 3 Quadrigons.

• These 3 component quadrigons of a Quadrangle are either
• * concave/concave/concave, or
• * convex/crossed/crossed.
• The first situation does not produce a real solution for circumscribed parabolas. So the best way to describe the positioning of 4 points producing circumscribed parabolas is by stating that these 4 points must not bound a concave form.
• This can be confirmed from the formulas.
• It contains the element √(-p q r (p+q+r)). Now (p+q+r) is always > 0 because barycentric coordinates represent areas and the sum of the 3 areas = area of the reference triangle. So √(-p q r (p+q+r)) is only real when p.q.r is negative. That is only when the 4^th point in a Quadrigon is a potential mirror point across the sides of the triangle and not across the vertices or inside the triangle as shown in next picture.

The construction of the 2 circumscribed parabolas was described by Newton. This method can be found at [7] as well as [14]. Calculation of the infinity points according to this method delivers these coordinates:

Parabola Equations CT-notation:

r (pq-pr+qr+q²) x y – q (pq-pr-qr-r²) x z – p (q+r)² y z + 2 Tc (r x y – q x z) = 0

Infinity points CT-notation:

( p (q + r) : –p q – Tc : –p r + Tc )

( p (q + r) : –p q + Tc : –p r – Tc )

where Tc = √(-p q r (p+q+r))

Parabola Equations DT-notation:

(p²-q²+r²) (q² x²-p² y²)+(p²+q²-r²) (r² x²-p² z²) + Td ((q²-r²) x²-p² (y²-z²)) = 0

Infinity points DT-notation:

(2 p² : -p²-q²+r² – Td : -p²+q²-r² + Td)

(2 p² : -p²-q²+r² + Td : -p²+q²-r² – Td)

where Td = √ ((p-q-r) (p+q-r) (p-q+r) (p+q+r))

The Parabola infinity points are equal to the infinity points of the Nine-point Conic.

Properties

• The sides of the Medial Triangle (of the QA-Diagonal Triangle) are tangential to both circumscribed Quadrangle Parabolas.
• The foci of both circumscribed Quadrangle Parabolas lie on the circumcircle of the Medial Triangle (MT) of the QA-Diagonal Triangle (= Nine-point Circle of the QA-Diagonal Triangle).
• The foci of both circumscribed Quadrangle Parabolas are concyclic with QA-P11 and the involutary conjugate of QA-P11.
• The directrices of both parabolas intersect in QA-P11 (the Circumcenter of the QA-Diagonal Triangle).
• The axes of both parabolas intersect in QA-P6 (the Parabola Axes Crosspoint) which is the Midpoint of QA-P2 (Euler-Poncelet Point) and QA-P4 (Isogonal Center).
• The axes of both parabolas are parallel to the asymptotes of the Nine-point Conic.
• The QA-Diagonal Triangle (QA-Tr1) is self-polar wrt QA-2Co1a and QA-2Co1b.

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