QA-Co-1: QA-DT-Conic Perspector


A circumscribed QA-DT-Conic is a conic through the vertices of the QA-Diagonal Triangle QA-Tr1.
There is a special property for these conics:
Let Sij be the intersection, other than the vertex of the QA-Diagonal Triangle, of the QA-DT-Conic and line Pi.Pj, for all combinations of (i,j) (1,2,3,4).  The lines Sij.Skl, for all combinations of (i,j,k,l) (1,2,3,4), concur in a new point which we shall call here the QA-DT-Conic-Perspector.

This subject was being developed by the specific observation of Angel Montesdeoca in QA-Ci1 and QA-P38. It was generalized by observations of Eckart Schmidt, Randy Hutson and the author of EQF (July, 2012).

QA-Co--1-QA-FT-Conic-Perspector01           
 
Since the QA-DT-Conic Perspector is derived wrt a QA-Diagonal Triangle the coordinates are best expressed in DT-coordinates. 
1st DT-coefficient:  
      u1 u2 (v2 w1 - v1 w2) (p2 r2 v12 v22 (u2 w1 - u1 w2)2 + p2 q2 w12 w22 (u2 v1 - u1 v2)2 - q2 r2 u12 u22 (v2 w1 - v1 w2)2)
where S1, S2, S3, Q1(u1:v1:w1) and Q2(u2:v2:w2) are the defining points of the QA-DT-Conic.

Properties:
  • When QA-DT-Conic = QA-Ci1, then the QA-DT-Conic-Perspector is QA-P38.
  • When QA-DT-Conic = QA-Co1, then the QA-DT-Conic-Perspector is QA-P1.
  • When QA-DT-Conic = QA-Co4, then the QA-DT-Conic-Perspector is a point on the line QA-P22.QA-P29. No further properties were found.
  • When QA-DT-Conic = QA-Co5, then the QA-DT-Conic-Perspector is a point on the line QA-P16.QA-P17. It is also the intersection of tangents to QA-Co5 at QA-P1 and QA-P20 and it lies on the line through QA-P22 and the center of QA-Co5. (Randy Hutson, July, 2012).
    1st DT-coordinate: p2 (p2 - q2 - r2) (p4 q2 - 2 p2 q4 + q6 + p4 r2 + 4 p2 q2 r2 - q4 r2 - 2 p2 r4 - q2 r4 + r6)

     

     

Plaats reactie