QG-2P1: Inscribed Square Centers
Each quadrigon has 2 Inscribed Squares.
The centers of these squares are QG-2P1a and QG-2P1b.
Construction:
The construction of these squares is described in Ref-14, Ref-15g, pages 12,13, Ref-15h pages 8,9, Ref-21 and Ref-29.
Another construction with the use of conics:
- Let CIR be the circle with diameter the points S2=P1.P2 ^ P3.P4 and S3=P1.P4 ^ P2.P3. The center of this circle is QG-P2.
- Now QG-2P1a and QG-2P1b are the intersection points of the line QG-P2.QA-P23 and the conic CON which is the Involutary Conjugate (QA-Tf2) of CIR. This conic is the locus of all inscribed Rhombi in the Reference Quadrigon (Eckart Schmidt, October 3, 2012).
Moreover:
- QA-P23 can be constructed as the Involutary Conjugate (QA-Tf2) of the Orthocenter of the QA-Diagonal Triangle.
- CON can be constructed as the conic through S2, S3 and the Harmonic Conjugates (see Ref-13) of the intersection point of CIR with Pi.Pj wrt (Pi, Pj).
- By reflecting a Quadrigon side wrt QG-2P1a/b, the reflected line will intersect the opposite Quadrigon side in one of the vertices of the inscribed Square (Eckart Schmidt, October 5, 2012). In this way the vertices of the Inscribed Quadrigon can be found.
Coordinates:
CT-coordinates QG-2P1a/b in 1st QA-Quadrigon:
(a2 (c2 p2 q2 + b2 p2 r2 + a2 q2 r2) + 2 a2 p q r (-p SA + q SB + r SC) - 8 p2 (q2 + r2) Δ2 :
(-c2 p2 q2 + b2 p2 r2 + a2 q2 r2) SC + 2 p q r2 SC2 + (2 (a2 q2 (p2 + r2) - b2 p2 ( q2 + r2) + 2 a2 q2 r (p + q)) - 4 p2 q r SA + 4 p q3 SB) Δ - 8 p q2 (q + r) Δ2 :
(c2 p2 q2 - b2 p2 r2 + a2 q2 r2) SB + 2 p q2 r SB2 + (-2 (c2 p2 q2 + (-a2 + c2 ) p2 r2 - a2 q2 r2) - 4 p q r (p SA - r SC)) Δ)
The coordinates of the 2nd point can be derived by changing the sign in the terms with Δ (not with Δ2).
CT-coordinates QG-2P1a/b in 1st QL-Quadrigon:
(-(a2 l2 + c2 n2) SB + 2 l n SB2 - 4 c2 m n SC + 4 SA (-a2 l m + m2 SC) - 2 ((a2 l2 - c2 n2) + 2 m n SA - 2 l m SC) Δ + 8 m (l + n) Δ2 :
2 m n SA2 + SA ((a2 l2 - c2 n2) - 2 l m SC) + 2 ((-b2 l2 + c2 l2 + c2 n2) + 2 m (l - n) SA - 2 l n SB) Δ :
- c2 (a2 l2 + c2 n2) - 4 m2 SA2 + 2 c2 l n SB + 4 SA (c2 m n - l m SB) + 8 l2 Δ2)
The coordinates of the 2nd point can be derived by changing the sign in the terms with Δ (not with Δ2).
DT-coordinates QG-2P1a/b in 1st QA-Quadrigon:
(p (S (p2 Sa - r2 Sc) + p r Sb b2) : -q2 Sb (r Sc - p S) : -r (-p2 S2 + r2 Sc2))
The coordinates of the 2nd point can be derived by changing the sign in the terms with S.
DT-coordinates QG-2P1a/b in 1st QL-Quadrigon:
(m2 n (n2 S (S + Sa) - l2 (S + Sc) Sc - l n Sb b2) : l n (l (S + Sc) - n (S + Sa)) (l2 a2 - n2 c2) : l m2 (-n2 Sa (S + Sa) + l2 S (S + Sc) - l n Sb b2))
The coordinates of the 2nd point can be derived by changing the sign in the terms with S.
Properties:
- QG-2P1a and QG-2P1b are collinear with QG-P2 and QA-P23 on QG-L6.
- (QG-2P1a, QG-2P1b) lie with (QG-P2, QA-P23) in harmonic position on QG-L6 (Eckart Schmidt, September 6, 2012).
- QG-2P1a and QG-2P1b lie center symmetric on a Conic which is the QA-Tf2-mapping of the circle with diameter the vertices on the baseline of the QA-Diagonal Triangle. The tangents at the centers of the Inscribed Squares to this conic are parallel to QG-L1 (Eckart Schmidt, September 6, 2012). The Center of this conic is the midpoint of QG-2P1a and QG-2P1b and the intersection point of lines QG-P2.QA-P23 and QG-P12.QA-P6.