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 [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.

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:

(W_a W_b W_c – (W_a – V_a) (W_b – V_b) (W_c – V_c) – b² c² (q + r)² V_a – a² c² (p + r)² V_b – a² b² (p + q)² V_c) Δ / (V_a V_b V_c)

where:

V_a = (W_a (2p + q + r) – a² (p + q)(p + r)) / (p + q + r)

V_b = (W_b (p + 2q + r) – b² (q + r)(q + p)) / (p + q + r)

V_c = (W_c (p + q + 2r) – c² (r + p)(r + q)) / (p + q + r)

W_a = b² (p + q) + c² (p + r)

W_b = c² (q + r) + a² (q + p)

W_c = 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 [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 2^nd 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 [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 [34], Eckart Schmidt, QFG-message #1666.

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