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:
CT-Coordinates QG-P4 in 3 QA-Quadrigons:
- ((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) )
CT-Coordinates QG-P4 in 3 QL-Quadrigons:
* ((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)
DT-Coordinates QG-P4 in 3 QA-Quadrigons:
- (-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)
DT-Coordinates QG-P4 in 3 QL-Quadrigons:
- (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) !
- 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.
- 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.