QL-P4: Miquel Circumcenter
The circumcenters of the 4 component triangles of the Reference Quadrilateral are concyclic on the Miquel Circle. It is special that QL-P1, the Miquel Point, also resides on the Miquel Circle. The circumcenter of this circle is QL-P4.
This point is mentioned at Ref-48 and Ref-49 by Morley in resp. 1900 and 1902 as Center of the Center Circle. Morley describes this point as a recursive point for n-Lines.
The point is also mentioned in 1921 by J.W. Clawson (see Ref-31). He referred to this point as the “center of the circumcentric quadrangle”.
a2 (a2 (m - n) (l - m) (n - l)
+ b2 ( l - m) (n2 + l m - 2 m n)
+ c2 (n - l ) (m2 + l n - 2 m n))
Sa2 (m2-n2)3 - Sb2 (l2-n2)2 (3 m2+n2) + Sc2 (l2-m2)2 (m2+3 n2)
+2(m2-n2) (Sa Sc (l2-m2)2 + Sa Sb (l2-n2)2 - S2 (l2 (l2+m2+n2)-3 m2 n2))
QL-P4 lies on these lines:
- QL-P4 is the Reflection of QL-P5 in QL-P6.
- QL-P4 is the Railway Watcher (see QL-L-1) of QL-L1 (Newton Line) and QL-L4 (Morley Line).
- QL-P4 is the Gergonne-Steiner Point (QA-P3) as well as the Isogonal Conjugate (QA-P4) as well as the Midray Homothetic Center (QA-P8) as well as the QA-DT-Orthocenter (QA-P12) from the Circumcenter Quadrangle (see QL-P3). These 4 QA-points concur because the Circumcenter Quadrangle is concyclic.
- QL-P4 is the Centroid of the 8 centers of circles as described in rule (9) from Steiner (seen as a system of 8 random points). See QL-8P1.
- The QA-Möbius Conjugate (QA-Tf4) of QL-P4 is a apoint on the line QA-P4.QG-P9.