QL-2P1: Plücker Pair of Points
Plücker proved that the circles having the three diagonals as diameters have two common points which lie on the line joining the four triangles' orthocenters (Wells 1991). That’s why these points are called the Plücker Points. See Ref-13, item “Complete Quadrilateral”.
These points have added value because the circle with QL-P5 (Clawson Center) as center passing through QL-P1 (Miquel Point) also passes through the Plücker Points.
Coordinates:
CT-Coordinates:
Different presentations of QL-2P1a: 1st Plücker Point:
Plu1a = {SB SC Ta : SC (Tn + (l - m) n TT) : SB (Tm + (l - n) m TT)}
Plu1b = {SC (Tn + (m - l) n TT) : SA SC Tb : SA (Tl + (m - n) l TT)}
Plu1c = {SB (Tm + (n - l) m TT) : SA (Tl + (n - m) l TT) : SA SB Tc}
Different presentations of QL-2P1b: 2nd Plücker Point:
Plu2a = {SB SC Ta : SC (Tn - (l - m) n TT) : SB (Tm – (l - n) m TT)}
Plu2b = {SC (Tn - (m - l) n TT) : SA SC Tb : SA (Tl – (m - n) l TT)}
Plu2c = {SB (Tm - (n - l) m TT) : SA (Tl + (n - m) l TT) : SA SB Tc}
where:
Ta = l m n (–a2 l + SB n + SC m)
Tb = l m n (+SA n – b2 m + SC l)
Tc = l m n (+SA m + SB l – c2 n)
Tl = +2 Δ2 ( l2 m2 + l m n2 + l2 n2 + l m2 n) - l m n (a2 SA l + b2 SB m + c2 SC n)
Tm = +2 Δ2 (m2 n2 + l2 m n + l2 m2 + l m n2) - l m n (a2 SA l + b2 SB m + c2 SC n)
Tn = +2 Δ2 ( l2 n2 + l m2 n + m2 n2 + l2 m n) - l m n (a2 SA l + b2 SB m + c2 SC n)
TT = +2 Δ√ [Δ2 (l m + l n + m n)2 - l m n (a2 SA l + b2 SB m + c2 SC n)]
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
Properties:
- The Plücker Points QL-2P1 a/b lie on the Steiner Line (QL-L2).
- The circle with center QL-P5 (Clawson Center) and passing through QL-P1 (Miquel Point) also passes through the Plücker Points QL-2P1 a/b.
- The Plücker Points QL-2P1 a/b are the Clawson-Schmidt Conjugates (QL-Tf1) of QL-2P4 a/b.