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.

enter image description here

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

Some new equilateral triangles I discovered recently in here:


Today I have just been found that the theorem above was found early by Dr. Floor van Lamoen. His paper Equilateral chordal triangles.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.