Question Number 116669 by Eric002 last updated on 05/Oct/20
Commented by Eric002 last updated on 05/Oct/20
$${d}\acute {{e}montrer}\:{que}\:\:{les}\:{triangles}\:{ABC}\:{et}\:{IJK} \\ $$$${ont}\:{le}\:{m}\hat {{e}me}\:{centre}\:{de}\:{gravit}\acute {{e}} \\ $$
Answered by mindispower last updated on 05/Oct/20
$${soit}\:{G}\:{le}\:{centr}\:{de}\:{ABC} \\ $$$${et}\:\:{G}'\:{celui}\:{de}\:{KIJ} \\ $$$${dans}\:{tous}\:{ce}\:{qui}?{suit} \\ $$$${AB}\:=\overset{\rightarrow} {{A}B} \\ $$$${GA}+{GB}+{GC}=\mathrm{0}…\mathrm{1} \\ $$$${G}'{K}+{G}'{I}+{G}'{J}=\mathrm{0}…\mathrm{2} \\ $$$$\mathrm{2}\Leftrightarrow \\ $$$${G}'{A}+{AK}+{G}'{B}+{BI}+{G}'{C}+{CJ}=\mathrm{0} \\ $$$$\mathrm{1}\Leftrightarrow{GG}'+{G}'{A}+{GG}'+{G}'{B}+{GG}'+{G}'{C}=\mathrm{0} \\ $$$$\Leftrightarrow\mathrm{3}{GG}'+{G}'{A}+{G}'{B}+{G}'{C}=\mathrm{0} \\ $$$${AK}={CA},{BI}={AB},{CJ}={BC} \\ $$$${AK}+{BI}+{CJ}={AB}+{CJ}+{BC} \\ $$$$\mathrm{2}\Leftrightarrow{G}'{A}+{G}'{B}+{G}'{C}+{CA}+{AB}+{BC}=\mathrm{0} \\ $$$$\mathrm{2}\Leftrightarrow{G}'{A}+{G}'{B}+{G}'{C}=\mathrm{0} \\ $$$$\mathrm{2}\Leftrightarrow\mathrm{3}{GG}'=\mathrm{0}\Rightarrow{G}={G}' \\ $$$$ \\ $$
Commented by Eric002 last updated on 05/Oct/20
$${bravo}\:{professeur} \\ $$
Commented by mindispower last updated on 06/Oct/20
$${je}\:{vous}\:{en}\:{prie} \\ $$$${je}\:{suis}\:{pas}\:{professeur}\:\:{just}\: \\ $$$${un}\:{simple}\:{matheux} \\ $$