**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 4
^{th}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:*

a

^{2}(a^{2}(l - m) (l - n) + b^{2}(2 m - n) (m - l) + c^{2}(2 n - m) (n - l))*Least Sum of 4 Square Distances in CT-notation:*

(-a

^{4}m^{2}n^{2}(l - m) (l - n) + b^{2}c^{2}l^{2}(m - n)^{2}((l + m + n) (2 l + m + n) - m n) - b

^{4}l^{2}n^{2}(m - l) (m - n) + c^{2}a^{2}m^{2}(n - l)^{2}((l + m + n) (l + 2 m + n) - n l ) - c

^{4}m^{2}l^{2}(n - l) (n - m) + a^{2}b^{2}n^{2}(l - m)^{2}((l + m + n) (l + m + 2 n) - l m))/ ((l + m + n)

^{2}(a^{2}m n (l - m) (l - n) + b^{2}n l (m - l) (m - n) + c^{2}l m (n - l) (n - m)))*1st DT-coordinate:*

m

^{2}n^{2 }(a^{4}(l^{2}- m^{2}) (l^{2}- n^{2}) - b^{4}(m^{2}- l^{2}) (m^{2}- n^{2}) - c^{4}(n^{2}- l^{2}) (n^{2}- m^{2})+ 2 a

^{2}b^{2}(l^{2}- m^{2}) n^{2}+ 2 a^{2}c^{2}m^{2}(l^{2}- n^{2}))*Least Sum of 4 Square Distances in DT-notation:*

(4 S

^{2}l^{2}m^{2}n^{2}(a^{2}l^{2}+ b^{2}m^{2}+ c^{2}n^{2}))/ (a

^{4}(l^{2}- m^{2}) (l^{2}- n^{2}) (l^{2}m^{2}+ l^{2}n^{2}- m^{2}n^{2}) + 2 b^{2}c^{2}l^{4}(m^{2}- n^{2})^{2}+ b

^{4}(m^{2}- n^{2}) (m^{2}- l^{2}) (l^{2}m^{2}- l^{2}n^{2}+ m^{2}n^{2}) + 2 c^{2}a^{2}m^{4}(n^{2}- l^{2})^{2}+ c

^{4}(n^{2}- l^{2}) (n^{2}- m^{2}) (-l^{2}m^{2}+ l^{2}n^{2}+ m^{2}n^{2}) + 2 a^{2}b^{2}n^{4}(l^{2}- m^{2})^{2})

*Properties:*- QL-P26 is collinear with QA-P42 and QG-P12.
- QL-P26 is the Clawson-Schmidt Conjugate of the 2
^{nd}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).
- 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.