Morley equilateral triangle is the nice theorem in Eulidean Geometry. I found an equilateral triangle and a group circle related to the Morley triangle and angle trisectors:

*Let $ABC$ be a triangle with angles $A, B, C$. Let points $D$ and $G$ be chosen on side $AB$, points $I$ and $F$ be chosen on side $BC$, points $E$ and $H$ be chosen on side $CA$ so that:*

$$\begin{cases} \angle EDA =\frac{2B}{3}+\frac{C}{3} \\ \angle FEC =\frac{A}{3}+\frac{2B}{3} \\ \angle GFB = \frac{A}{3}+\frac{2C}{3} \\ \angle HGA =\frac{B}{3}+\frac{2C}{3} \\ \angle IHC = \frac{2A}{3}+\frac{B}{3} \end{cases}$$

*1. Then six points $D$, $E$, $F$, $G$, $H$, $I$ lie on a circle and $\angle DIB = \frac{2A}{3}+\frac{C}{3}$*

*2. Let $HI \cap FG \equiv A_1$, $DE\cap HI \equiv B_1$, $FG \cap DE \equiv C_1$ then $A_1B_1C_1$ be an equilateral triangle. Two triangles $A_1B_1C_1$ and $ABC$ are perspective.*

*3. The triangle $A_1B_1C_1$ and the Morley triangle are homothetic.*

My question:Which is the barycentric coordinate of the perspector in item 2?

Some new equilateral triangles I discovered recently in here: