QLP12: QLCentroid or Lateral Centroid
QLP12 is the Centroid of the six intersection points of the 4 defining QLLines as mentioned by J.W. Clawson. See Ref31, page 41.
It is called the Lateral Centroid because it is constructed in the Quadrilateral environment.
It can be constructed as the midpoint of any set of two triangles formed from the six intersection points.
Just like in the QAenvironment, the Centroid in the QLenvironment is the Midpoint of 1^{st} and 2^{nd} Quasi Centroids.
Other ways of construction:
 QLP12 is the Tripolar Centroid ^{*)} of the QACentroids in the 3 QLQuadrigons.
 Let Gi be the Centroid of component triangle Lj.Lk.Ll and let Gjkl be the Centroid of Centroids Triangle Gj.Gk.Gl , where (i,j,k,l) ∈ (1,2,3,4). Lines Gi.Gjkl concur in QLP12.
 Let Gi be the Triangle Centroid (X2) of Triangle Lj.Lk.Ll. Let TCi be the Tripolar Centroid *) of the 3 intersection points of Lj, Lk, Ll with Li. Now all lines Gi.TCi (i=1,2,3,4) concur in QLP12.
 QLP12 is the homothetic Center of the Reference Quadrilateral L1.L2.L3.L4 and the homothetic Quadrilateral formed by the lines parallel to the lines L1, L2, L3, L4 through the Centroids of the corresponding component triangles.
*) The Tripolar Centroid is the Centroid of a “flat” triangle formed by 3 collinear points.
Constructionmethod: Suppose P, Q, R are collinear points not on the line at infinity. Let M = Midpoint(Q, R).
The segment PM has two trisectors. The trisector closer to M is the Tripolar Centroid.
Coordinates:
1st CTcoordinate:
(m  n) (l (l + m + n)  3 (l m + l n  m n)) (note that this formula is independent of a,b,c)
1st DTcoordinate:
(m^{2}n^{2}) (m^{2} (l^{2}n^{2}) + n^{2} (l^{2}m^{2}))
Properties:
 QLP12 lies on these lines:
 QLP12 is the Midpoint of QLP14 (1^{st} QLQuasi Centroid) and QLP15 (2^{nd} QLQuasi Centroid).
 QLP12 is the Midpoint of QLP8 and QLP18.
 The distance ratios between points QLP20, QLP22, QLP12, QLP5 are 3 : 1 : 2.
 d(QLP6 , QLP12) = d(QLP2 , QLP12) / 2. (d = distance)
 d(QLP5 , QLP12) = d(QLP2 , QLP3) / 3.
 QLP12 is the point where the sum of the squares of the distances to the 6 intersection points of the Reference Quadrilateral is minimal.
 QAP26 is the Centroid of the Triangle formed by the 3 QLversions of QLP12 (note Eckart Schmidt).
 QLP12 is the Centroid of the 3 Triangles being the 3 QLversions of QLP8, QGP4, QAP10 (all Centroid related points).
 QLP12 is the Centroid of the Triangle formed by the 3 QLpoints QLP2, QLP4, QLP5 (J.W. Clawson, see Ref31).
 QLP12 is the Centroid of QLTr2.

Let X2i (i=1,2,3,4) be the circumcenters of the triangles (Lj,Lk,Ll), where j,k,l=different numbers from (1,2,3,4) unequal i. Let L2i be the lines through X2i parallel to Li. QLP12 is the Homothetic Center of the Reference Quadrilateral and (L21,L22, L23, L24). This construction is similar to the construction of QLP20 and QLP22.