Question and Answers Forum

All Questions      Topic List

Geometry Questions

Previous in All Question      Next in All Question      

Previous in Geometry      Next in Geometry      

Question Number 151294 by Jonathanwaweh last updated on 19/Aug/21

Answered by ArielVyny last updated on 20/Aug/21

(a+b+c)=1→a^2 +a(b+c)+b(a+c)+b^2 +c^2 +c(b+a) a^2 +b^2 +c^2 +2ab+2ac+2bc=1 A=(1/2)ac.sin(b^� )=(1/2)bc.sin(a^� )=(1/2)ab.sin(c^� ) c^2 =a^2 +b^2 −2a.b.cos(c^� ) a^2 =c^2 +b^2 −2c.b.cos(a^� ) →alkashi′s theorem b^2 =a^2 +c^2 −2a.c.cos(b^� ) a^2 +b^2 +c^2 =2a^2 +2b^2 +2c^2 −2(a.b.cos(c^� )+c.b.cos(a^� )+a.c.cos(b^� )) a^2 +b^2 +c^2 >0 2a^2 +2b^2 +2c^2 −2(a.b.cos(c^� )+c.b.cos(a^� )+a.c.cos(b^� ))=1−2(ab+ac+bc)>0 par identification 2a^2 +2b^2 +2c^2 <1→a^2 +b^2 +c^2 <(1/2) (∗)

(a+b+c)=1→a2+a(b+c)+b(a+c)+b2+c2+c(b+a)a2+b2+c2+2ab+2ac+2bc=1A=12ac.sin(b^)=12bc.sin(a^)=12ab.sin(c^)c2=a2+b2−2a.b.cos(c^)a2=c2+b2−2c.b.cos(a^)→alkashi′stheoremb2=a2+c2−2a.c.cos(b^)a2+b2+c2=2a2+2b2+2c2−2(a.b.cos(c^)+c.b.cos(a^)+a.c.cos(b^))a2+b2+c2>02a2+2b2+2c2−2(a.b.cos(c^)+c.b.cos(a^)+a.c.cos(b^))=1−2(ab+ac+bc)>0paridentification2a2+2b2+2c2<1→a2+b2+c2<12(∗)

Terms of Service

Privacy Policy

Contact: info@tinkutara.com