QA-P34: Euler-Poncelet Point of the Centroid Quadrangle

QA-P34 is the Euler-Poncelet Point of the quadrangle formed by the Centroids of the Component Triangles of the Reference Quadrangle.
The Euler-Poncelet point also can be constructed in similar Quadrangles:
• QA-P1 also is Euler-Poncelet Point of the Nine-point Center Quadrangle.
• QA-P2 also is Euler-Poncelet Point of the Orthocenter Quadrangle.
• QA-P3 also is Euler-Poncelet Point of the Circumcenter Quadrangle.

Surprisingly now QA-P1, QA-P2, QA-P3 and QA-P34 have mutual distance ratios similar to the corresponding points in the Triangle Environment on the Euler Line.
This point was contributed by Eckart Schmidt (12/18/2011).

Coordinates:
1st CT-Coordinate:
2 a4 q r (p + q) (p + r) (2 p + q + r)
+ b4 p r (p + q) (q + r) (3 p + q + r)
+ c4 p q (p + r) (q + r) (3 p + q + r)
+ b2 c2 p (q + r) ((p + q) (q + r) (r + p) - 3 q r (2 p + q + r))
- a2 b2 p r (p + q) (3 q (p + q) + (4 q + r) (p + r))
- a2 c2 p q (p + r) (3 r (p + r) + (4 r + q) (p + q))

1st DT-Coordinate:
((p2 + q2 + r2)2 - 4 (p4 + q2 r2)) (-b2 p2 + a2 q2) (c2 p2 - a2 r2)
+ 4 p2 (-p2 + q2 + r2)(-c2 q2 + b2 r2) ((-b2 p2 + a2 q2) + (c2 p2 - a2 r2))

Properties:
• QA-P34 lies on these QA-lines:
QA-P1.QA-P2             (-1 : 4)
QA-P5.QA-P29          ( 2 :  1)
QA-P20.QA-P35        (5 : -2)

Plaats reactie