3f:null 40:["$","title",null,{"children":"Triangle centers | skydog23 | Observable"}] 46:T524,texmd`Here we discuss how to represent the classical triangle centers within 2D EPGA. Let $A$, $B$, and $C$ be the normalized vertices of the triangle. Then define the three sides: $$a:=B\vee C,~~~b:=C\vee A,~~~c:=A\vee B$$ Note that these sides are **not** normalized. The centroid is the intersection of the medians. These are the join of the vertex with the midpoint of the opposite side: $$m_a:=a \vee (B+C),~~~m_b:=b \vee (C+A),~~~m_c:=c \vee (A+B)$$ **Exercise:** The centroid is $M_c := m_a\wedge m_b \cong A+B+C$. The orthocenter is simple to obtain since the altitudes of the triangle can be simply obtained via the inner product of a vertex with its opposite side: $$h_a:=a\cdot A,~~~h_b:=b \cdot B,~~~h_c:=c \cdot C$$ The orthocenter is then $O_c := h_a \vee h_b$. The circumcenter arises in a similar way. Instead of using the perpendicular to the side through the opposite vertex, we use the perpendicular to the side through the midpoint of the side: $$p_a:=a\cdot (B+C),~~~p_b:=b \cdot (C+A),~~~p_c:=c \cdot (A+B)$$ The circumcenter is then $C_c := p_a \vee p_b$. The demo below shows these three centers, along with their common line, the *Euler line* of the triangle. It's well-known that $M_c$ lies between $C_c$ and $O_c$, and that $O_c$ lies twice as far from $M_c$ as does $C_c$. There `47:Ta18,Algebra({ metric:[0,1,1], basis:['1','e0','e1','e2','e01','e02','e12','e012'] }, ()=>{ // The Euler line connects several important points in any non equilateral triangle. // Not many identities needed.. var point = (x,y)=>1e12+x*1e02+y*1e01, {sin, cos, PI} = Math; // Three points. var A=point(-.3,0.5).Normalized, B=point(0.5,.5).Normalized, C=point(0,0).Normalized, // The edges a=()=>B&C,b=()=>C&A,c=()=>A&B, // The centroid centroid = ()=>(A+B+C).Normalized, // The orthocenter oa=()=>(a<(b<(c<oa^ob, // The circumcenter ca=()=>(c<<(A+B)).Normalized, cb=()=>(a<<(B+C)).Normalized, cc=()=>(b<<(C+A)).Normalized, circum=()=>ca^cb, // The Euler line euler = ()=>(ortho&circum).Normalized, orthoL = ()=>polarLine(ortho).Normalized, // symmetric product: appears equal to polar line of orthocenter // which is furthermore perpendicular to the euler line s = ()=>a*b*c+b*c*a+c*a*b+a*c*b+b*a*c+c*b*a; function polarLine(X){ var ax = ()=> X&A, bx =()=> X&B, cx =()=> X&C; // joining lines to three corners var AX = ()=> ax^a, BX =()=> bx^b, CX = ()=> cx^c; // intersection points with three sides var am = ()=> BX&CX, bm =()=> CX&AX, cm =()=> AX&BX; var AM = ()=> am^a, BM = ()=> bm^b, CM = ()=> cm^c; return ()=>AM&BM; }; // The hyperbolic disk var disc = [...Array(100)] .map((x,i)=>point(sin(i/50*PI),cos(i/50*PI))) .map((x,i,a)=>[x,a[i+1]||a[0]]); return this.graph(()=>{ var odote = euler<orthoL^euler,"X",s,"s"] }, { // more render properties for the default items. pointRadius:1.5, // point radius lineWidth:2, // line width fontSize:1.5, // font size grid:true, // grid scale:2, width:window.innerWidth, animate:true}); })48:[] 44:{"id":"aae28310d46cd1ab","slug":"triangle-centers","title":"Triangle centers","thumbnail":"81d89c4477d462bb896d2b1cc1411299f65b44fdf83f5cecaf4166c091a7f42e","access_level":"public","create_time":"$D2020-03-05T07:12:08.309Z","update_time":"$D2020-07-20T17:30:37.808Z","view_time":null,"first_public_time":"$D2020-07-20T17:31:55.050Z","has_unpublished_edits":null,"fork":false,"stars":2,"starred":false,"owner":{"type":"team","id":"c12ededdbf97bde4","login":"skydog23","name":"","avatar_url":"https://avatars-next.observableusercontent.com/57d55b329e0de79c4f1a8c1447c6cb3689b05343c2cde413e6c55225ab16ea26","tier":"starter_2024","bio":"","following":false},"version":69,"publish_version":69,"role":null,"authored":false,"authors":[{"type":"individual","id":"8ebee8d27ef26f03","login":"skydog23","name":"","avatar_url":"https://avatars-next.observableusercontent.com/57d55b329e0de79c4f1a8c1447c6cb3689b05343c2cde413e6c55225ab16ea26","tier":null,"bio":"","following":false}],"members":[{"type":"individual","id":"8ebee8d27ef26f03","login":"skydog23","name":"","avatar_url":"https://avatars-next.observableusercontent.com/57d55b329e0de79c4f1a8c1447c6cb3689b05343c2cde413e6c55225ab16ea26","tier":null,"role":"editor"}],"comments":[],"custom_thumbnail":false,"trashed":false,"lapsed":null,"subscribed":null,"forks":5,"collection":null,"type":"notebook","body":{"titling":"auto","title":"Triangle centers","cells":[{"id":0,"value":"md`# Triangle centers`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":46,"value":"$46","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":2,"value":"$47","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":5,"value":"Algebra = require('ganja.js')","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":49,"value":"import {texmd} from '@kelleyvanevert/katex-within-markdown'","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"}],"files":"$W48","stdlib":"1"}} 45:"$44:body"