QL-P1: Miquel Point
The Miquel Point is the common point of the circumcircles of the 4 component triangles of the Reference Quadrilateral.
This point is also called the Steiner Point or Clifford Point.
He describes 134 circles of “more or less interest” through this point.
Coordinates:
1st CT-coordinate:
a2 m n/(m - n)
1st DT-coordinate:
b2/(l2-n2) - c2/(l2-m2)
Properties:
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The Miquel Point QL-P1 lies on these lines:
The collinearity of QG-P1, QA-P3 and QL-P1 is discussed at Ref-34, QFG-messages #2987, #2989, #2990, #2992, #3000, #3009, #3011, #3027, #3029 and proven synthetically at Ref-60, page 150 (O ~ QA-P3, R ~ QG-P1, Mr ~ QL-P1).
– QL-Ci3 (Miquel Circle),
– QL-Ci5 (Plücker Circle),
– QL-Ci6 (Dimidium Circle).
- QL-P1 lies on QL-Ci2 (Nine-point Circle of the QL-Diagonal Triangle). See Ref-32 as well as Ref-11 (Hyacinthos Message # 12896 from Quang Tuan Bui).
- QL-P1 lies on QL-Cu1 (QL-Quasi Isogonal Cubic).
- QL-P1 lies on QL-Qu1 (QL-Cardiode).
- QL-P1 is the isogonal conjugate of the Infinity Point of the Newton line with respect to each of the four component triangles (see Ref-4 page 41).
- The symmetric lines of the Steiner Line QL-L2 in the 4 quadrilateral lines coincide at QL-P1.
- QL-P1 relates pairwise to all present line segments in the quadrilateral. It is the center of similarity of line segments Li^Lk . Li^Ll and Lj^Lk . Lj^Ll, where (i,j,k,l) ∊ (1,2,3,4). As a consequence triangles QL-P1 . Li^Lk . Li^Ll and QL-P1 . Lj^Lk . Lj^Ll are similar (see also Ref-9).
- QL-P1 is the perspector of the QL-Diagonal Triangle and the Triangle formed by the 3 QL-versions of QA-P3 (Gergonne-Steiner Point).
- The Clawson-Schmidt Conjugate (QL-Tf1) is “centered” around QL-P1.
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The Miquel Point QL-P1 of a Quadrilateral L1.L2.L3.L4 is the Isogonal Conjugate of the Infinity Point of the Trilinear Polar of the Isotomic Conjugate of the Trilinear Pole of Li wrt Triangle Lj.Lk.Ll, where (i,j,k,l) is any combination of {1,2,3,4}.
In short: MP = Ic(Ip(TPolar(It(TPole(Li))))).
(Francisco Javier García Capitán, Hyacinthos message #21271(Ref-11), November 5, 2012)
- QL-P1 is the QA-Tf2 image (Involutary Conjugate) of QG-P16 (Eckart Schmidt, November 26, 2012).
- QL-P1 lies on the Polars (see Ref-13, Polar) of QL-P2, QL-P7, QL-P9 wrt QL-Co1.
- Let ABCD be a Component Quadrigon of the Reference Quadrilateral and let P be its Miquel Point. If m is the distance between the midpoints of the diagonals of ABCD, then PA/(AB*AD) = PB/(BA*BC)=PC/(CB*CD)=PD/(DA*DC)=1/(2*m) (see Ref-11, Hyacinthos message # 9086).
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Let Ti = Triangle formed by the 3 lines Lij, Lik, Lil ((i,j,k,l) ∈ (1,2,3,4)) where Lmn = Line perpendicular to Ln at intersection point Lm^Ln. The 4 triangles Ti (i=1,2,3,4) have one common point, being QL-P1. Moreover the circumcenters of the 4 triangles Ti (i=1,2,3,4) lie on QL-Ci3. Found by Antreas P. Hatzipolakis, see Ref-33, Anopolis messages # 466,467.
- The QA-Orthopole (QA-Tf3) of QL-P1 is a point on the Newton Line QL-L1.
- The QA-Möbius Conjugate (QA-Tf4) of QL-P1 is QG-P5.
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QL-P1 lies on the polar of QG-P16 wrt QG-Co2 (and invers) (Eckart Schmidt, October 9, 2013). See Ref-34, QFG # 286.
- The QA-Triple Triangle of QL-P1 is Perspective and Cyclologic wrt all QA-Component Triangles (Ref-34, QFG#977 by Seiichi Kirikami). See QA-Tr-1.
- The QA-Triple Triangle of QL-P1 is Orthologic wrt the Triple Triangles of QG-P7, QG-P9 and QL-P6. See QA-Tr-1.
- The reflections of QL-P1 in the 4 basic lines of the Reference Quadrilateral lie on QL-L2. See Ref-34, Seiichi Kirikami, QFG message # 1091.
- The reflections of QL-L2 in the 4 basic lines of the Reference Quadrilateral concur in QL-P1. See Ref-34, Seiichi Kirikami, QFG message # 1095.
- The line through the intersection point of Li and Lj (i,j=1,2,3,4) and through the intersection point of the reflected lines of QL-L3 in Li and Lj passes through QL-P1. See Ref-34, Seiichi Kirikami, QFG message # 1096.
- The Orthopole of ANY line through QL-P1 wrt ANY Component Triangle of the Reference Quadrilateral lies on QL-L3. See Ref-33, Anopolis # 637.