QG-P4: 1st QG-Quasi Centroid

The 1st QG-Quasi Centroid is the Diagonal Crosspoint of the X2-Quadrigon.
The X2-Quadrigon is defined by its vertices being the Triangle Centroids of the component triangles of the Reference Quadrigon.
QG-P4 is actually the center of mass of the Quadrigon when it is convex and when its surface is being made of some evenly distributed material.
This point and other 1st QG-Quasi points are described in Ref-5.

Coordinates:
• ((p + r)2 + q r            : (p + r) (p + 2q + r) : (p + r)2 + p q )
• ((q + r) (2p + q + r) : (q + r)2 + p r            : (q + r)2 + p q )
• ((p + q)2 + q r            : (p + q)2 + p r          : (p + q) (p + q + 2r) )
* ((lm+ln+mn)2-lmn(l+4m+4n)    :    (n-l)(lm2+l2n+m2n-2lm(-l+m+n))   :   (m-l)(ln2+l2m+mn2)-2ln(-l+m+n))
* ((n-m)(l2m+m2n+l n-2lm(l-m+n))    :    (lm+ln+mn)2-lmn(4l+m+4n)    :   (l-m)(lm2+mn2+ln2-2mn(l-m+n)))
* ((m-n)(l2m+l2n+mn2-2ln(l+m-n))    :    (l-n)(lm2+m2n+ln2-2mn(l+m-n))    :    (lm+ln+mn)2-lmn(4l+4m+n))
CT-Area of QG-P4-Triangle in the QA-environment:
• 2 p q r Δ / (9 (p + q) (p + r) (q + r))                                           (equals 1/9 * area QA-Diagonal Triangle)

CT-Area of QG-P4-Triangle in the QL-environment:
• 4 l2 m2 n2 Δ /(9 (-lm+ln+mn) (lm+ln-mn) (lm-ln+mn))      (equals 1/9 * area QL-Diagonal Triangle)

• (-4 p2 ( p2-q2-r2)   :   (p2-r2)2 -2 q2 (-p2+q2-r2)- q4    :  -4 r2 (-p2-q2+r2))
• ((r2-q2)2 -2 p2 (-r2+p2-q2)- p4   :   -4 q2 (-r2-p2+q2)  :  -4 r2 (  r2-p2-q2))
• (-4 p2 (-q2-r2+p2) :    -4 q2 (q2-r2-p2)   :   (q2-p2)2 - 2 r2 (-q2+r2-p2)- r4)
• (2 m2 (m2-n2)   : -m2 (l2+m2+n2)+3 l2 n2  : 2 m2 (-l2+m2))
• (- l2 (l2+m2+n2)+3 n2 m2   :   2 l2 (-n2+l2) : 2  l2  (  l2-m2))
• (2 n2 (-m2+n2) :  2 n2 (n2-l2)   :   -n2 (l2+m2+n2)+3 m2 l2)

DT-Area of QG-P4-Triangle in the QA-environment:
• S / 18                                                                                (equals 1/9 * area QA-Diagonal Triangle)

DT-Area of QG-P4-Triangle in the QL-environment:
• S / 18                                                                                            (equals 1/9 * area QL-Diagonal Triangle)

Properties:
• QG-P4, QG-P5, QG-P6, QG-P7 are collinear on QG-L4, the 1st QG-Quasi Euler Line.
• QG-P4 is the Reflection of QG-P1 in QG-P8.
• QG-P4 is the Reflection of QG-P8 in QA-P1 (QA-Centroid).
• QG-P4 is the Reflection of QA-P10 (Centroid QA-DT) in QL-P12 (QL-Centroid) (Eckart Schmidt, July, 2012) !
• QG-P4 is the Centroid of the 1st QG-Quasi Diagonal Triangle: QG-Tr1.
• QG-P4 is the Centroid of the Triangle formed by QA-P5 and the two vertices of the QA-Diagonal Triangle unequal QG-P1.
• QA-P25 is the Centroid of the triangle formed by the 3 QA-versions of QG-P4.
• QL-P14 is the Centroid of the triangle formed by the 3 QL-versions of QG-P4.
• Divide the sides of a Quadrigon into three equal parts. The figure formed by connecting and extending adjacent points on either side of the Quadrigon form a parallelogram, Wittenbauer's Parallelogram. See Ref-13. QG-P4 is the center of this parallelogram.
• QA-P34 lies on the circle defined by the 3 QA-versions of QG-P4 (note Eckart Schmidt).
• The triangle formed by the 3 QA-versions of QG-P4 is homothetic and perspective with the QA-Diagonal Triangle. The side lengths are 1/3 of the side lengths of the QA-Diagonal triangle. Their Perspector is QA-P1.
• The area of the triangle formed by the 3 QA-versions of QG-P4 equals 1/9 * the area of the QA-Diagonal Triangle.
• The area of the triangle formed by the 3 QL-versions of QG-P4 also equals 1/9 * the area of the QL-Diagonal Triangle. However both triangles are not homothetic neither perspective.

Vernieuwen