An new equilateral triangle related to the Morley triangle

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: