QL-Ci5: Plücker Circle

The Plücker Circle is the circle through the Plücker Points (QL-2P1a and QL-2P1b) and the Miquel Point (QL-P1). Its center is the Clawson Center (QL-P5).

Equation in CT-notation:
c2 x y + b2 x z + a2 y z + (x + y + z) ((R - T1/T4) x + (R - T2/T4) y + (R - T3/T4) z) = 0

where:
T1 = + TM2 c2 ( l - n)2 + TN2 b2 (l - m)2 - 2 TM TN SA (l - m) (l – n)
T2 = + TN2 a2 (m - l)2 + TL2 c2 (m - n)2 - 2 TN TL SB (m - n)(m - l)
T3 = + TL2 b2 (n - m)2 + TM2 a2 (n - l)2 - 2 TL TM SC (n - l)(n - m)
T4 = 1024 Δ4 (l - m)2 (m - n)2 (n - l)2
R = R13 R22 - 8 R1 R2 R3 Δ2 - 64 R1 R4 Δ4                                                           (R = radius2 Plücker Circle)
R1 = a2 mn(l-m)(l-n) + b2 nl(m-l)(m-n) + c2 lm(n-l)(n-m)
R2 = a2 l SA + b2 m SB + c2 n SC
R3 =           SL (2 SA (a2 m2 n2 + b2 l2 n2 + c2 l2 m2) + 4 l m n (b2 SB n + c2 SC m - b2 c2 l) - 16 Δ2 m2 n2)
+ SM (2 SB (a2 m2 n2 + b2 l2 n2 + c2 l2 m2) + 4 l m n (c2 SC l + a2 SA n - a2 c2 m) - 16 Δ2  l2  n2)
+ SN (2 SC (a2 m2 n2 + b2 l2 n2 + c2 l2 m2) + 4 l m n (a2 SA m + b2 SB l - a2 b2 n) - 16 Δ2 l2 m2)
R4 =    - b2 c2 SL2 - c2 a2 SM2 - a2 b2 SN2 + 2 a2 SA SN SM + 2 b2 SB SN SL + 2 c2 SC SL SM
and:
Δ = Area = 1/4[(a + b + c) (–a + b + c) (a – b + c) (a + b – c)]
SA = (–a2 + b2 + c2) / 2                          SB = (+a2 – b2 + c2) / 2                         SC = (+a2 + b2 – c2) / 2
SL = l (m - n)2 (-mn + lm + ln)            SM = m (n - l)2 (mn + lm - ln)              SN = n (l - m)2 (mn + lm + ln)
TL = -2 l R2 +16Δ2 (-mn + lm + ln)     TM =-2 m R2 +16Δ2 (mn + lm - ln)    TN =-2 n R2 +16Δ2 (mn - lm + ln)

Properties:
• QL-Ci5 is orthogonal wrt the QL-Diagonal Triangle Circumcircle (QL-Ci1),  just like all 3 Plücker Diagonal Circles are.
• The QA-Orthopole (QA-Tf3) of QL-Ci5 is a circle through QL-P5.
• The inverse of QL-P1 wrt QL-Ci1 lies on QL-Ci5 (Bernard Keizer, July 27, 2014). See Ref-34, QFG message # 643.
• Given quadrilateral (L1,L2,L3,L4). Let X1 be the reflections of orthopole of L1 wrt triangle (L2,L3,L4) through L1. Similarly, we have X2,X3,X4. Then X1,X2,X3,X4 are concyclic on QL-Ci5. See Ref-34, Tran Quang Hung, QFG#2081,#2083.