## 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]

## Basic Symmetries in Nine-Point Circle

### Personal reading report

Motivation Someone on Discord asked about the existence of the nine-point circle. It’s well-known that that can be proved by homothety. Little reminder about homothety Homothety preserves angles (and thus parallel lines). Homothetic polygons are similar, so the ratio of the corresponding sides is the same. Considering the radii of a circle under a homothety, we see that a homothety maps a circle to another circle. Notation H: orthocenter G: centroid O: circumcenter ω: circumcircle HA: feet of altitude with respect to A. [Read More]