QL-P26: Least Squares Point

QL-P26 is the point in a quadrilateral such that the sum of the squares of the distances to the 4 basic lines is minimal.  Construction 1:
This construction is a modified version of the construction of Coolidge as described in Ref-25. Let O (origin), A and B be random non-collinear points.
Go = Quadrangle Centroid of the projection points of O on the 4 basic lines of the Reference Quadrilateral.
Ga = Quadrangle Centroid of the projection points of O on the 4 lines through point A parallel to the 4 basic lines of the Reference Quadrilateral.
Gb = Quadrangle Centroid of the projection points of O on the 4 lines through point B parallel to the 4 basic lines of the Reference Quadrilateral.
Let Sa = Ga.Go ^ O.Gb and Sb = Gb.Go ^ O.Ga.
Construct A1 on line O.A such that Sa.Ga : Ga.Go = O.A : A.A1.
Construct B1 on line O.B such that Sb.Gb : Gb.Go = O.B : B.B1.
Construct P such that O.A1.P.B1 is a parallelogram and where O and P are opposite vertices.
P is the Least Squares Point QL-P26.

Construction 2:
Here follows an alternative construction of QL-P26 derived from the property found by Eckart Schmidt (October 5, 2012) that in a Quadrigon QL-P26 is the Centroid of its pedal Quadrangle:
• Let P1, P2, P3 P4 be the intersection points of L1, L2, L3, L4 in some QL-Quadrigon.
• Let G1a and G1b be Centroids of the pedal Quadrangles of random points  Q1a, Q1b on the perpendicular through P1 to P3.P4.  Let T1 = Q1a.Q1b ^ G1a.G1b.
• Let G2a and G2b be Centroids of the pedal Quadrangles of random points Q2a, Q2b on the perpendicular through P2 to P3.P4. Let T2 = Q2a.Q2b ^ G2a.G2b.
• Let G3a and G3b be Centroids of the pedal Quadrangles of random points Q3a, Q3b on the perpendicular through P3 to P1.P2. Let T3 = Q3a.Q3b ^ G3a.G3b.
• Let G4a and G4b be Centroids of the pedal Quadrangles of random points Q4a, Q4b on the perpendicular through P4 to P1.P2. Let T4 = Q4a.Q4b ^ G4a.G4b.
• Now QL-P26 = T1.T2 ^ T3.T4.

Construction 3:
There is another remarkable construction of QL-P26 described by M. Léon Ripert. See Ref-45, pages 110-112 and Ref-34, QFG #457.
• Let L1.L2.L3 be a reference triangle with 4th Line L4 crossing the 3 Lines of the reference triangle.
• Q4 = L1 ^ ((L2^L3)-Symmedian of triangle L2.L3.L4)
R4 = L2 ^ ((L3^L1)-Symmedian of triangle L3.L1.L4)
S4 = L3 ^ ((L1^L2)-Symmedian of triangle L1.L2.L4)
• Q4, R4, S4 are collinear on a line SL4.
• For the other 3 Component triangles corresponding lines SL1, SL2, SL3 can be constructed.
• These 4 Lines concur in QL-P26.

Coordinates:
1st CT-coordinate:
a2 (a2 (l - m) (l - n) + b2 (2 m - n) (m - l) + c2 (2 n - m) (n - l))
Least Sum of 4 Square Distances in CT-notation:
(-a4 m2 n2 (l - m) (l - n) + b2 c2 l2 (m - n)2 ((l + m + n) (2 l + m + n) - m n)
- b4 l2 n2 (m - l) (m - n) + c2 a2 m2 (n - l)2 ((l + m + n) (l + 2 m + n) - n l )
- c4 m2 l2 (n - l) (n - m) + a2 b2 n2 (l - m)2 ((l + m + n) (l + m + 2 n) - l m))/
((l + m + n)2 (a2 m n (l - m) (l - n) + b2 n l (m - l) (m - n) + c2 l m (n - l) (n - m)))

1st DT-coordinate:
m2 n2 (a4 (l2 - m2) (l2 - n2) - b4 (m2 - l2) (m2 - n2) - c4 (n2 - l2) (n2 - m2)
+ 2 a2 b2 (l2 - m2) n2 + 2 a2 c2 m2 (l2 - n2))
Least Sum of 4 Square Distances in DT-notation:
(4 S2 l2 m2 n2 (a2 l2 + b2 m2 + c2 n2))
/ (a4 (l2 - m2) (l2 - n2) (l2 m2 + l2 n2 - m2 n2) + 2 b2 c2 l4 (m2 - n2)2
+ b4 (m2 - n2) (m2 - l2) (l2 m2 - l2 n2 + m2 n2) + 2 c2 a2 m4 (n2 - l2)2
+ c4 (n2 - l2) (n2 - m2) (-l2 m2 + l2 n2 + m2 n2) + 2 a2 b2 n4 (l2 - m2)2)

Properties:
• QL-P26 is collinear with QA-P42 and QG-P12.
• QL-P26 is the Clawson-Schmidt Conjugate of the 2nd intersection point of the line QL-P1.QL-P13 with the Dimidium Circle QL-Ci6.
• QL-P26 lies on the line QL-P1.CSC(QL-P13).   (CSC = Clawson-Schmidt Conjugate, see QL-Tf1)
• QL-P26 lies on the line CSC(QL-P17).CSC(QL-P24), which is the perpendicular bisector of QL-P1.CSC(QL-P6).
• The three QL-versions of QG-P16 (Schmidt Point) are collinear on former line, which is also the QL-Tf1 image of the Dimidium circle QL-Ci6 (Eckart Schmidt, November 30,2012).
• QL-P26 is the Centroid of its pedal Quadrangle (Eckart Schmidt, October 5, 2012).
• QL-P26 lies on the orthogonal hyperbola through QL-P1, QL-P19 and the 3 intersection points QL-P1.QA-P4 ^ QL-P19.QA-P6 (constructing per QL-Quadrigon the related QA-P4 point and QA-P6 point) (Eckart Schmidt, October 5, 2012).
• In a quadrilateral, the three QL-versions of QA-P2 (Euler) and QA-P4 (isogonic) produce inversely similar triangles. The fixed point of this similarity is QL-P26. See Ref-34, QFG-message #1430 of Benedetto Scimemi). With this affine transformation the 3 versions of angle QA-P2i.QL-P26.QA-P4i (i=a,b,c) share the same angle bisectors (additional note of Bendetto).
• Every Component Triangles has a conic Coi (i=1,2,3,4) through its vertices, centroid and its perspector with the QL-Diagonal Triangle. The Isogonal conjugate of this conic wrt this Component Triangle is a line ILi (i=1,2,3,4). The 4 lines ILi (i=1,2,3,4) are concurrent in QL-P26. See Ref-34, QFG-messages #457 and #1458 of Bernard Keizer.

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