## Second Homothety between Nine-Point Circle and Circumcircle

### Proof of Euler line by h(G, −1/2)

Motivation I saw someone illustrating his/her solution with a “superior triangle”. This reminds me the homothety about the centroid of factor −1/2. The above picture \usetikzlibrary{calc} for computing coordinates from those of existing points. (A)!.25!(B) means $(A)+.25[(B)-(A)]$. \begin{tikzpicture}[scale=2] \coordinate (D) at (-0.7,1); \coordinate (E) at (-1,0); \coordinate (F) at (1,0); \coordinate (A) at ($(E)!.5!(F)$); \coordinate (B) at ($(F)!.5!(D)$); \coordinate (C) at ($(D)!.5!(E)$); \coordinate (G) at ($(D)!.5!(E)!1/3!(F)$); \draw (A) -- (B) -- (C) -- cycle; \draw (D) -- (E) -- (F) -- cycle; \begin{scriptsize} \fill (G) circle (0. [Read More]