nine-point circle on Solarized Sublime Sekai
https://vincenttam.gitlab.io/tags/nine-point-circle/
Recent content in nine-point circle on Solarized Sublime SekaiHugo -- gohugo.ioenMon, 06 Jun 2022 04:36:17 +0200Second Homothety between Nine-Point Circle and Circumcircle
https://vincenttam.gitlab.io/post/2022-06-06-second-homothety-between-nine-point-circle-and-circumcircle/
Mon, 06 Jun 2022 04:36:17 +0200https://vincenttam.gitlab.io/post/2022-06-06-second-homothety-between-nine-point-circle-and-circumcircle/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.Basic Symmetries in Nine-Point Circle
https://vincenttam.gitlab.io/post/2022-06-04-basic-symmetries-in-nine-point-circle/
Sat, 04 Jun 2022 22:29:26 +0200https://vincenttam.gitlab.io/post/2022-06-04-basic-symmetries-in-nine-point-circle/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.