Prove that $MA^{2}+MB^{2}+MC^{2}geq frac{4S_{ABC}}{sqrt{3}}(1+frac{OM^{2}}{3R^{2}})$
Triangle $ABC$ has a circumcircle $(O;R)$. M is a point in triangle $ABC$. Prove that $MA^{2}+MB^{2}+MC^{2}geq frac{4S_{ABC}}{sqrt{3}}(1+frac{OM^{2}}{3R^{2}})$
When i prove , i think lots of way to start. I see $frac{OM^{2}}{R^{2}}$, i think we can use euler theorem about pedal triangle. With MD,ME,MF is perpendicular to BC,CA,AB we have $S_{DEF} = frac{1}{4}(1-frac{OM^{2}}{R^{2}}).S_{ABC}$
Beside that, i also think we can use the lemma:$ frac{MA}{a}+frac{MB}{b}+frac{MC}{c}geq sqrt{3}$
Can anyone help me pls?
geometry euclidean-geometry
asked Nov 29 '18 at 15:24
Trong TuanTrong Tuan
1268
add a comment |
Triangle $ABC$ has a circumcircle $(O;R)$. M is a point in triangle $ABC$. Prove that $MA^{2}+MB^{2}+MC^{2}geq frac{4S_{ABC}}{sqrt{3}}(1+frac{OM^{2}}{3R^{2}})$
When i prove , i think lots of way to start. I see $frac{OM^{2}}{R^{2}}$, i think we can use euler theorem about pedal triangle. With MD,ME,MF is perpendicular to BC,CA,AB we have $S_{DEF} = frac{1}{4}(1-frac{OM^{2}}{R^{2}}).S_{ABC}$
Beside that, i also think we can use the lemma:$ frac{MA}{a}+frac{MB}{b}+frac{MC}{c}geq sqrt{3}$
Can anyone help me pls?
geometry euclidean-geometry
asked Nov 29 '18 at 15:24
Trong TuanTrong Tuan
1268
add a comment |
Triangle $ABC$ has a circumcircle $(O;R)$. M is a point in triangle $ABC$. Prove that $MA^{2}+MB^{2}+MC^{2}geq frac{4S_{ABC}}{sqrt{3}}(1+frac{OM^{2}}{3R^{2}})$
When i prove , i think lots of way to start. I see $frac{OM^{2}}{R^{2}}$, i think we can use euler theorem about pedal triangle. With MD,ME,MF is perpendicular to BC,CA,AB we have $S_{DEF} = frac{1}{4}(1-frac{OM^{2}}{R^{2}}).S_{ABC}$
Beside that, i also think we can use the lemma:$ frac{MA}{a}+frac{MB}{b}+frac{MC}{c}geq sqrt{3}$
Can anyone help me pls?
geometry euclidean-geometry
asked Nov 29 '18 at 15:24
Trong TuanTrong Tuan
1268
Triangle $ABC$ has a circumcircle $(O;R)$. M is a point in triangle $ABC$. Prove that $MA^{2}+MB^{2}+MC^{2}geq frac{4S_{ABC}}{sqrt{3}}(1+frac{OM^{2}}{3R^{2}})$
When i prove , i think lots of way to start. I see $frac{OM^{2}}{R^{2}}$, i think we can use euler theorem about pedal triangle. With MD,ME,MF is perpendicular to BC,CA,AB we have $S_{DEF} = frac{1}{4}(1-frac{OM^{2}}{R^{2}}).S_{ABC}$
Beside that, i also think we can use the lemma:$ frac{MA}{a}+frac{MB}{b}+frac{MC}{c}geq sqrt{3}$
Can anyone help me pls?
geometry euclidean-geometry
geometry euclidean-geometry
asked Nov 29 '18 at 15:24
Trong TuanTrong Tuan
1268
asked Nov 29 '18 at 15:24
Trong TuanTrong Tuan
1268
asked Nov 29 '18 at 15:24
Trong TuanTrong Tuan
1268
asked Nov 29 '18 at 15:24
Trong TuanTrong Tuan
1268
asked Nov 29 '18 at 15:24
Trong TuanTrong Tuan
1268
1268
add a comment |
add a comment |
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