QA-Cu-1: Circumscribed QA-Cubics

In EQF (Encyclopedia of Quadri-Figures) 3 types of Quadrangle Cubics are mentioned.

QA-Cubic Type 1
QA-Cubic Type 1 is a cubic that can be constructed as follows:
Let P1, P2, P3, P4 be the vertices of the Reference Quadrangle.
Let V (u:v:w) be a variable point.
Let Lv be some line through V.
Let IC(Lv) be the Involutary Conjugate (QA-Tf2) of Line Lv.
IC(Lv) is a conic since QA-Tf2 is a transformation of the 2nd degree.
The locus of the intersection IC(Lv) ^ Lv is a QA-Cubic Type 1.
Because of its function V is called the Pivot Point of the QA-Cubic Type 1.
Examples: QA-Cu1QA-Cu5, where resp. V = QA-P4, QA-P5, QA-P10, QA-P19, QA-P1.
The general equation in CT-notation is:
q r x2 (v z - w y) + p r y2 (w x - u z) + p q z2 ( u y - v x) = 0
The general equation in DT-notation is:
p2 y z (v z - w y) + q2 x z (w x - u z) + r2 x y (u y - v x) = 0

This QA-Cubic Type 1 has some interesting general properties:
• The tangents at P1, P2, P3, P4 are concurrent in the Pivot Point V on the cubic.
• The tangents at DT1, DT2, DT3 are concurrent in a point W on the cubic, which is the Involutary Conjugate of V.
• The vertices of the Cevian Triangle of Pivot Point V wrt the QA-Diagonal Triangle QA-Tr1 lie on the cubic.
• V.W is the only line through V for which Q1 is Double Point itself wrt the created QA-Line Involution, whilst W is the 2nd Double Point on this line.
V.W is tangent in V at the cubic and also at the conic (P1,P2,P3,P4,V)
These cubics can all be seen as “pivotal isocubics” like described by Bernard Gibert (Ref-17b). The reference system here is not a triangle but a quadrangle. The Isoconjugation here is the Involutary Conjugation. Point V is the pivot.
Cubic QA-Cu1 is also a circular cubic because the imaginary circular infinity points lies on this cubic.

QA-Cubic Type 2

QA-Cubic Type 1 is a cubic that can be constructed as follows:
Let P1, P2, P3, P4 be the vertices of the Reference Quadrangle.
Let V (u:v:w) be a variable point.
Let Lv be some line through V.
Let IC(Lv) be the Involutary Conjugate (QA-Tf2) of Line Lv.
IC(Lv) is a conic since QA-Tf2 is a transformation of the 2nd degree.
The locus of the intersection of IC (Lv) ^ the perpendicular at V to Lv is a QA-Cubic Type-2.
Example: QA-Cu7 (QA-Quasi Isogonal Cubic) with V = QA-P4.
Cubic QA-Cu7 is also a circular cubic because the imaginary circular infinity points lies on this cubic.
The general equation in CT-notation is:
(-a2 v w + c2 v (v + w) + b2 w (v + w )) (q r x3 - p r x2 y - p q x2 z)
+ (-b2 u w + c2 u (u + w) + a2 w (u + w)) (p r y3 - p q y2 z - q r x y2)
+ (-c2 u v + b2 u (u + v) + a2 v (u + v )) (p q z3 - q r x z2 - p r y z2)
+ (a2 (q r u2 - p r u v + q r u v - p r v2 - p q u w + q r u w + p q v w + p r v w + 2 q r v w - p q w2)
+ b2 (-q r u2 + p r u v - q r u v + p r v2 + p q u w + 2 p r u w + q r u w - p q v w + p r v w - p q w2)
+ c2 (-q r u2 + 2 p q u v + p r u v + q r u v - p r v2 + p q u w - q r u w + p q v w - p r v w + p q w2)) x y z = 0
The general equation in DT-notation is:
(-Sa u2 + Sb v (u + w) + Sc (u + v) w) (r2 y2 + q2 z2) x
+(Sa u (v + w) - Sb v2 + Sc (u + v) w) (r2 x2 + p2 z2) y
+(Sa u (v + w) + Sb v (u + w) - Sc w2) (q2 x2 + p2 y2) z
-(r2 (b2 u2 + 2 Sc u v + a2 v2) + q2 (c2 u2 + 2 Sb u w + a2 w2) + p2 (c2 v2 + 2 Sa v w + b2 w2)) x y z = 0

QA-Cubic Type 3
QA-Cubic Type 3 is a cubic that can be constructed as follows:
Let P1, P2, P3, P4 be the vertices of the Reference Quadrangle.
Let V (u:v:w) be a variable point.
Let Lv be some line through V.
Let IC(Lv) be the Involutary Conjugate (QA-Tf2) of Line Lv.
IC(Lv) is a conic since QA-Tf2 is a transformation of the 2nd degree.
The intersection of IC(Lv) with Lv results in 2 intersection points.
The locus of the Midpoint of these 2 intersection points (which is the Involution Center of the QA-Line Involution on line Lv) produces a QA-Cubic Type 3.
Example: QA-Cu6, where V = QA-P1.
The general equation in CT-notation is:
r w ((p u + q u + p w) y - (p v + q v + q w) x) x y
+ q v ((r v + p w + r w) x - (p u + r u + p v) z) x z
+ p u ((q u + q v + r v) z - (r u + q w + r w) y) y z
+ (q r u (w - v) + p r v (u - w) + p q w (v - u)) x y z = 0
The general equation in DT-notation is:
p2 (w y - v z) (-2 u y z + v z (x - y + z) + w y (x + y - z))
+q2 (u z - w x) (u z (-x + y + z) - 2 v x z + w x (x + y - z))
+r2  (v x - u y) (u y (-x + y + z) + v x (x - y + z) - 2 w x y) = 0

Properties:
• The vertices of the Reference Quadrangle lie on this cubic.
• The intersection point of the line through V parallel to the side Pi.Pj intersected with the opposite side Pk.Pl is a point on this cubic for all instances of (i,j,k,l) ∊ (1,2,3,4).

Notes from Bernard Gibert (2012, January 22):
Type 1 cubics are pK(P x Q, P) wrt ABC, in particular :
QA-Cu1 = pK(P x igP, P), circular cubic see CL035 and SITP 4.2.1
QA-Cu2 = pK(W1, P), central cubic see CL017 and SITP 3.1
QA-Cu3 = pK(P x X2P, P), where X2P is the centroid of the cevian triangle of P, see CL007
QA-Cu4 = pK(cP x ccP x ct(P2),P), seems complicated...
QA-Cu5 = pK(W4, P), a K+, see CL017 and CL049
Type 2 cubics are nK isocubics wrt the cevian triangle of P. See SITP 1.5.3.
it is the locus of M such that P/M lies on the tangent at Q to the circle with center M passing through Q.
QA-Cu7 is obtained with Q = igP. It is a focal cubic with focus P/igP, passing through the center S of the rectangular circum-hyperbola through P and P/S which is the infinite real point.
Type 3 cubics are spK(P, Midpoint PQ) cubics with pole P x Q. See CL055.
These are nodal cubics with node Q.
QA-Cu6 is obtained with Q = ccP. It is a K+.
notations : c, g, t, i mean complement, isogonal, isotomic, inverse as usual.
SITP = Special Isocubics...