**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”.

*Coordinates:**1st CT-coordinate:*

a

^{2}(a^{2 }(m - n) (l - m) (n - l)+ b

^{2}( l - m) (n^{2}+ l m - 2 m n)+ c

^{2}(n - l ) (m^{2}+ l n - 2 m n))*1st DT-coordinate:*

Sa

^{2}(m^{2}-n^{2})^{3}- Sb^{2}(l^{2}-n^{2})^{2}(3 m^{2}+n^{2}) + Sc^{2}(l^{2}-m^{2})^{2}(m^{2}+3 n^{2})+2(m

^{2}-n^{2}) (Sa Sc (l^{2}-m^{2})^{2}+ Sa Sb (l^{2}-n^{2})^{2}- S^{2 }(l^{2}(l^{2}+m^{2}+n^{2})-3 m^{2}n^{2}))

*Properties:*-
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.