QA-P38: Montesdeoca-Hutson Point

QA-P38 is the Perspector of the Reference Quadrangle with the Cyclocevian Quadrangle.
The Cyclocevian Quadrangle CC1.CC2.CC3.CC4 is defined by:
CCi = Cyclocevian Conjugate of Pi wrt Pj.Pk.Pl for all combinations of (i,j,k,l)  (1,2,3,4).
The definition of Cyclocevian Conjugate can be found at Ref-13.
This point was separately found by Angel Montesdeoca (Hyacinthos message 21075, see Ref-11) and Randy Hutson (private mail to author EQF) in the same week (June, 2012).

Construction
(by Angel Montesdeoca):
• Let S1.S2.S3 be the QA-Diagonal Triangle of the Reference Quadrangle P1.P2.P3.P4.
• For each vertex Pi, we take the triangle TjTkTl, where Tj the intersection of the sidelines PkPl with circumcircle of the triangle S1.S2.S3 (other than S1, S2, S3).
• Qi = Perspector of the triangle PjPkPl and TjTkTl.
• QA-P38 is the common intersection point of lines Pi.Qi.

Coordinates:

1st CT-Coordinate:
a2 q r (p + q)(p + r) (-2 p + q + r) + b2 p r (p + q) (q + r) (2 p - q + r) + c2 p q (p + r) (q + r) (2 p + q - r)
-------------------------------------------------------------------------------------------------------------------------
-a2 q r (p + q)(p + r)                          + b2 p r (p + q) (q + r)                        + c2 p q (p + r) (q + r)
1st DT-Coordinate:
a2 (-c4 p2 q2 - b4 p2 r2 + a4 q2 r2)

Properties:
where S = the intersection point QA-P1.QA-P6 ^ QA-P22.QA-P29.
S is also the Involution Center of the line QA-P1.QA-P6 wrt the Reference Quadrangle.
• QA-P38.QA-P11 // QA-P1.QA-P32 // QA-P2.QA-P4 // QA-P7.QA-P8 // QA-P12.QA-P24.
• QA-P38 = Perspector CycloCevian Quadrangle CC1.CC2.CC3.CC4 with Reference Quadrangle (definition Randy Hutson). The Perspectrix (Perspective Axis) of P1.P2.P3.P4 and CC1.CC2.CC3.CC4 is the perpendicular bisector of QA-P2.QA-P4 (QA-L9), which happens to be the Involutary Conjugate (QA-Tf2) of the Circumcircle of the QA-Diagonal Triangle (QA-Tr1).
• QA-P38 = TCC-Perspector Pi wrt QA-Diagonal Triangle (i=1,2,3,4)   (note Angel Montesdeoca). Definition of the TCC-Perspector can be found in ETC at Ref-12 in the description just before X(1601).
• Let Sij = 2nd intersection point of the line Pi.Pj and the circumcircle of the QA-Diagonal Triangle, where (i,j) ∈ (1,2,3,4).  Now QA-P38 = S12.S34 ^ S13.S24 ^ S14.S23 (note Angel Montesdeoca). This is a general property not only for QA-Ci1 wrt the 6 sides of a Quadrangle, but also for each QA-DT-conic wrt the 6 sides of a Quadrangle. See QA-Co-1.

Vernieuwen