QA-Tr2: Miquel Triangle

The QL-Miquel Points (QL-P1) of the 3 Quadrigons of the Reference Quadrangle form a triangle Mi1.Mi2.Mi3.
This Triangle is described in Ref-15c “Eckart-Schmidt - Das Steiner Dreieck von vier Punkten”. In this paper the triangle is called the “Steiner Dreieck”.
Special is that the QA-versions of the Steiner Axes (see QL-Tf1) have the role of the internal and external angle bisectors in the Miquel Triangle.
                       
QA-Tr2-Miquel-Triangle-00

Coordinates/Areas:
3 QA-versions of Miquel Points in CT-notation:
                (a2(p + q) (p + r) - p (b2(p + q) + c2(p + r)) : -b2(p + q) (p + q + r) : -c2(p + r) (p + q + r))
                (a2(p + q) (p + q + r) : a2 q (p + q) - (-c2 q + b2 (p + q)) (q + r) : c2 (q + r) (p + q + r))
                (a2(p + r) (p + q + r) : b2 (q + r) (p + q + r) : -c2 (p + r) (q + r) + r (a2(p + r) + b2 (q + r)))
Area Miquel Triangle in CT-notation:
(Wa Wb Wc - (Wa - Va) (Wb - Vb) (Wc - Vc)
- b2 c2 (q + r)2 Va - a2 c2 (p + r)2 Vb - a2 b2 (p + q)2 Vc) Δ / (Va Vb Vc)
where:
Va = (Wa (2p + q + r) - a2 (p + q)(p + r)) / (p + q + r)
Vb = (Wb (p + 2q + r) - b2 (q + r)(q + p)) / (p + q + r)
Vc = (Wc (p + q + 2r) - c2 (r + p)(r + q)) / (p + q + r)
Wa = b2 (p + q) + c2 (p + r)
Wb = c2 (q + r) + a2 (q + p)
Wc = a2 (r + p) + b2 (r + q)
 
3 QA-versions of Miquel Points in DT-notation:
(-2 p2 (p2 Sa + q2 Sb + r2 Sc) :
a2 q2 (p2 - q2 + r2) + b2 p2 (-p2 + q2 + r2) - 2 c2 p2 q2 :
a2 r2 (p2 + q2 - r2) - 2 b2 p2 r2 + c2 p2 (-p2 + q2 + r2))
(a2 q2 (p2 - q2 + r2) + b2 p2 (-p2 + q2 + r2) - 2 c2 p2 q2 :
-2 q2 (p2 Sa + q2 Sb + r2 Sc) :
-2 a2 q2 r2 + b2 r2 (p2 + q2 - r2) + c2 q2 (p2 - q2 + r2))
(a2 r2 (p2 + q2 - r2) - 2 b2 p2 r2 + c2 p2 (-p2 + q2 + r2) :
-2 a2 q2 r2 + b2 r2 (p2 + q2 - r2) + c2 q2 (p2 - q2 + r2) :
-2 r2 (p2 Sa + q2 Sb + r2 Sc))

Properties:
  • QA-Tr2 and QA-Tr1 (QA-Diagonal Triangle) are perspective triangles with Perspector QA-P3 (Gergonne-Steiner Point).
  • QA-Tr2 is perspective with all 4 QA-Component Triangles. These pairs of triangles are also cyclologic related with Cyclologic Centers QA-P4 and another point. See Ref-34, QFG#976, #977.
  • QA-P3 and QA-P4 are mutually isogonal conjugates wrt the Miquel Triangle.
  • The vertices of QA-Tr2 lie on the cubic QA-Cu1.
  • QA-P9 lies on the circumcircle of the Miquel Triangle.
  • The circumcenter of the Miquel Triangle lies on the line through the 3 QA-versions of QL-P5 (Clawson Center).
  • The intersection point of the QA-Cu1 cubic and its asymptote lies on the circumcircle of the Miquel Triangle opposite to QA-P9 (note Eckart Schmidt).
  • In a Quadrigon QA-Tr2 is perspective with the Triangle formed by QA-P4 and the vertices of the QA-Diagonal Triangle (QA-Tr1) unequal QG-P1, with perspector QG-P16 (Eckart Schmidt, November 26, 2012).
  • The vertices of QA-Tr2 can be constructed as the 2nd intersection point of the circles (QA-P4,Pi,Pj) and (QA-P4,Pk,Pl), where (i,j,k,l) = (1,2,3,4) / (1,3,2,4) / (1,4,2,3).
  • All QA-Tr2-circumconics through the intersection of QA-Cu1 and its asymptote cut QA-Cu1 in two QA-Tr2-isogonal conjugated further points. See Ref-34, Eckart Schmidt, QFG-message #1666.
  • All pivotal isogonal isocubics wrt QA-Tr2 intersect QA-Cu1 in two QA-Tr2-isogonal conjugated points collinear with the pivot. See Ref-34, Eckart Schmidt, QFG-message #1666.

 


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