# Generalization of Tucker circle, Conway circle and van Lamoen circle

Theorem 9.1 in this paper as follows is a generalization of Turker circle. Turker circles is a generalization of many circles as: Cosine Circle, circum circle, First Lemoine Circle, Gallatly Circle, Kenmotu Circle, Taylor Circle, Apollonius Circle.

The Conway circles and Floor van Lamoen circle are also special case of Turker circle.

I also given some anothert nice special cases of Theorem 9.1, example Theorem 9.2, Theorem 9.3, Theorem 9.6, Theorem 9.7 of that paper.

Theorem 9.1. Let $$ABC$$ be a triangle, let points $$D$$, $$G$$ be chosen on side $$AB$$, points $$I$$, $$F$$ be chosen on side $$BC$$, points $$E$$, $$H$$ be chosen on side $$CA$$, let $$k$$, $$l$$ are real number such that:

$$\begin{cases} \angle EDA =kA+lB+(1-k-l)C\\ \angle FEC =(1-l)A+(k+l)B-kC\\ \angle GFB = (1-k-l)A+kB+lC\\ \angle HGA =-kA+(1-l)B+(k+l)C \\ \angle IHC=lA+(1-k-l)B+kC \end{cases}$$

Then six points $$D, E, F, G, H, I$$ lie on a circle and $$\angle DIB = (k+l)A-kB+(1-l)C$$ Coverse of Theorem 9.1: Let $$ABC$$ be a triangle, let points $$D$$, $$G$$ be chosen on side $$AB$$, points $$I$$, $$F$$ be chosen on side $$BC$$, points $$E$$, $$H$$ be chosen on side $$CA$$ and six points $$D, E, F, G, H, I$$ lie on a circle. Then exist two real numbers $$k$$, $$l$$ such that:

$$\begin{cases} \angle EDA =kA+lB+(1-k-l)C\\ \angle FEC =(1-l)A+(k+l)B-kC\\ \angle GFB = (1-k-l)A+kB+lC\\ \angle HGA =-kA+(1-l)B+(k+l)C \\ \angle IHC=lA+(1-k-l)B+kC \\ \angle DIB = (k+l)A-kB+(1-l)C \end{cases}$$

My question: Is the converse of Theorem 9.1 true?