Der Luis,

> 1) X(110) of ABC is a Miquel point of the complete quadrangle bounded by the sidelines of ABC and its Lemoine axis. References?, previous research?

Nice result!

I only think you should say that X(110) of ABC is the Miquel point of the complete QUADRILATERAL bounded by the sidelines of ABC and its Lemoine axis.

I don't know about earlier research regarding a complete quadrilateral derived from a reference triangle and a special line wrt to this reference triangle.

However there has been done some research by Randy Hutson and Eckart Schmidt regarding a complete quadrangle derived from a reference triangle and a special point wrt to this reference triangle.

I did do some quick rearch of myself:

* X(5) of ABC is the Morley Point of the complete QUADRILATERAL bounded by the sidelines of ABC and its Lemoine axis.

(Morley Point see:

http://www.chrisvantienhoven.nl/quadrilateral-objects/17-mathematics/quadrilateral-objects/104-ql-p2.html )

* X(6) of ABC is the Least Squares point of the complete QUADRILATERAL bounded by the sidelines of ABC and its EULER LINE.

(Least Squares Point see:

http://chrisvantienhoven.nl/quadrilateral-objects/17-mathematics/quadrilateral-objects/131-ql-p26.html )

* X(691) of ABC is the Miquel point of the complete QUADRILATERAL bounded by the sidelines of ABC and its EULER LINE.

* Also new Triangle Centers can be found this way. There is a new point that is the Miquel Circumcenter of the complete QUADRILATERAL bounded by the sidelines of ABC and its Lemoine axis

I think there is a lot more that can be found in this way.

Barycentric coordinates: a^4 (b^2 - c^2) (4 SA^2 - b^2 c^2)

Search: -6.120759622

It is a point on the Lemoine Axis and {3,690}, {32,2491}, {526,1511}, {878,3455}.

(Miquel Circumcenter see:

http://www.chrisvantienhoven.nl/quadrilateral-objects/17-mathematics/quadrilateral-objects/107-ql-p4.html )

Best regards,

Chris van Tienhoven