QA-P3: Gergonne-Steiner Point

QA-P3 has two main functions:
  1. Center of the Gergonne Steiner Conic QA-Co3,
  2. Common point of the four midray circles.
This point is called after Gergonne and Steiner because in the “Annales de Gergonne” the question was posed for the Quadrangle Conic with least eccentricity. Steiner solved this problem. The center of this conic (QA-Co3) happens to be QA-P3.
In Heinrich Dörrie’s book: “100 Great Problems of Elementary Mathematics” (see Ref-7), there is problem 54 ”The most Nearly Circular Ellipse Circumscribing a Quadrilateral”. This most Nearly Circular Ellipse is QA-Co3 with center QA-P3. Dörrie also refers to Steiner solving this problem in “Gergonne’s Annales de Mathematique”, t. XVII, p.284 (Crelle’s Journal, vol. II), as well as “Steiner Gesammelte Werke”, vol. I. See Ref-61.
It is also the common point of the 4 circles defined by the midpoints of Pi.Pj, Pi.Pk, Pi.Pk for all combinations of (i,j,k,l) ∈ (1,2,3,4). This common circle point is described shortly without name in Ref-2c: Jean-Louis Ayme, Le Point- d’Euler-Poncelet d‘un Quadrilatère on page 10. According to Jean-Louis Ayme this point was mentioned in the writing of Igor Federovitch Sharygin “Problemas de geometria”.
It is strongly related to QA-P2 (the Euler-Poncelet Point) in construction with circles of the same size as well as in position within the Quadrangle.
QA-P3 always appears at the “opposite side” of the reference quadrangle than QA-P2The QA-Centroid QA-P1 is their midpoint.
1st CT-Coordinate:
(a2 (p + q) (p + r) - b2 p (p + q) - c2 p (p + r))  *  (a2 q r (2 p + q + r) - b2 p r (q + r) - c2 p q (q + r))
1st DT-Coordinate:
               1  /  (-2 a2 q2 r2 + b2 r2 (p2 + q2 - r2) + c2 q2 (p2 - q2 + r2))

  • QA-P3 lies on these lines:
        QA-P1.QA-P2              (-1 : 2 => QA-P3 = Reflection of QA-P2 in QA-P1)
        QA-P20.QA-P29         (2 : -1 => QA-P3 = Reflection of QA-P20 in QA-P3)
        QA-P22.QA-P35         (5 : -4)
The collinearity of QG-P1, QA-P3 and QL-P1 is discussed at Ref-34, QFG-messages #2987, #2989, #2990, #2992, #3000, #3009, #3011, #3027, #3029 and proven synthetically at Ref-60, page 150 (O ~ QA-P3, R ~ QG-P1, Mr ~ QL-P1).
  • The Reflection of Pi (i=1,2,3,4) in QA-P3 is a point on the circumcircle of Pj.Pk.Pl.
  • QA-P3 is the Euler-Poncelet Point (QA-P2) of the 1st Circumcenter Quadrangle (note Eckart Schmidt).
  • QA-P3 is the Homothetic Center of the Antigonal Quadrangle and the 1st Centroid Quadrangle (the Antigonal of a point X isthe isogonal conjugate of the inverse in the circumcircle of the isogonal conjugate of X, see Ref-17a).
  • The Antigonal Quadrangle of the 1st Centroid Quadrangle is perspective to the Reference Quadrangle at QA-P3 (Randy Hutson, August, 2012).
  • QA-P3 is the Perspector of the QA-Diagonal Triangle (QA-Tr1) and the Triangle formed by the Miquel points of the 3 Quadrigons of the Reference Quadrangle (QA-Tr2). See Ref-15 where QA-P3 is called the “Z-Punkt” by Eckart Schmidt.
  • Let P1.P2.P3.P4 be the Reference Quadrangle. Let O1 be the circumcenter of Component Triangle P2.P3.P4. The reflections  of O1.QA-P4 in the sidelines of P2.P3.P4 bound a triangle that is perspective with P2.P3.P4 with perspector J1. This point is lying on the circumcircle of P2.P3.P4 and moreover is incenter or excenter of P2.P3.P4. Similarly J2, J3, J4 are constructed. P1.P2.P3.P4 and J1.J2.J3.J4 are perspective. Their perspector is QA-P3. See [34], QFG messages #609, #624, #625, #627.
  • QA-P3 is the Isogonal Conjugate of the Complement of QA-P4 wrt the QA-Diagonal Triangle QA-Tr1. See Ref-15f theorem 25.
  • QA-P3 is the Isogonal Conjugate of the AntiComplement of QA-P28 wrt the QA-Diagonal Triangle QA-Tr1.
  • QA-P3 is the Isogonal Conjugate of QA-P4 wrt the Miquel Triangle QA-Tr2 (Ref-15c page 5).
  • QA-P3 lies on the circumcircle of the triangle formed by the 3 QA-versions of QG-P5, QG-P15 as well as QG-P17.
  • QA-P3 lies on the circumcircle of the triangle formed by the 3 QA-versions of the 2nd Quasi-De Longchamps Points (not registered QG-point).
  • QA-P3 is the common intersection point of the 3 QA-versions of QL-Ci6, the Dimidium Circle (note Eckart Schmidt).
  • QA-P3 is the Antipode of QA-P2 in the Nine-point Conic (QA-Co1).
  • QA-P3 is the Antipode of QA-P20 in the QA-DT-P3-P12 Orthogonal Hyperbola (QA-Co4).
  • QA-P3 lies on the QA-DT-P4 Cubic (QA-Cu1).
  • QA-P3 is the Clawson-Schmidt Conjugate (QL-Tf1) of QG-P16 (Eckart Schmidt, November 26, 2012).
  • The QA-Orthopole (QA-Tf3) of QA-P3 is QA-P1.
  • The QA-Möbius Conjugate (QA-Tf4) of QA-P3 is QA-P9.
  • QA-P3 is the Cyclologic Center of the QG-P15-Triple Triangle wrt the QG-P1-Triple Triangle. See Ref-34, QFG #965, #966.
  • QA-P3 is the Orthology Center of the Triple Triangles of QG-P1/QL-P1 wrt the Triple triangles of QG-P7/QG-P9/QL-P6. See QA-Tr-1.
  • QA-P3 is the Perspector of the Triple Triangles (see QA-Tr-1) of these pairs {QG-P1, QL-P21}, {QG-P13, QL-P17}.
  • QA-P3 is the Parallelologic Center of the QG-P5 Triple Triangle wrt the Triple Triangles of QG-P7/QG-P9/QL-P6 TT. See QA-Tr-1.
  • The QA-Tf2-conic of the QA-Cu1-asymptote contacts QA-Cu1 in QA-P3. See Ref-34, Eckart Schmidt, QFG-message #1666.

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