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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$

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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?

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