• Đặt \(S_{MBC}=S_1;S_{MAC}=S_2;S_{MAB}=S_3\)
• Dựng \(AH\perp BC\text{ và }MK\perp BC\)
⇒ AH // MK
\(\Rightarrow\dfrac{AD}{MD}=\dfrac{AH}{MK}=\dfrac{\dfrac{1}{2}\times AH\times BC}{\dfrac{1}{2}\times MK\times BC}=\dfrac{S_{ABC}}{S_1}\)
\(\Rightarrow\dfrac{AM}{MD}=\dfrac{AD}{MD}-1=\dfrac{S_{ABC}}{S_1}-1=\dfrac{S_2+S_3}{S_1}\)
\(\Rightarrow\sqrt{\dfrac{AM}{MD}}=\sqrt{\dfrac{S_2+S_3}{S_1}}\)
• Tương tự, ta cũng có: \(\sqrt{\dfrac{BM}{ME}}=\sqrt{\dfrac{S_1+S_3}{S_2}};\sqrt{\dfrac{CM}{MF}}=\sqrt{\dfrac{S_1+S_2}{S_3}}\)
• Áp dụng bất đẳng thức AM - GM, ta có:
\(P=\sqrt{\dfrac{S_2+S_3}{S_1}}+\sqrt{\dfrac{S_1+S_3}{S_2}}+\sqrt{\dfrac{S_2+S_1}{S_3}}\)
\(\ge3\sqrt[6]{\dfrac{S_2+S_3}{S_1}\times\dfrac{S_1+S_3}{S_2}\times\dfrac{S_2+S_1}{S_3}}\)
\(\ge3\sqrt[6]{\dfrac{2\sqrt{S_2S_3}}{S_1}\times\dfrac{2\sqrt{S_1S_3}}{S_2}\times\dfrac{2\sqrt{S_2S_1}}{S_3}}=3\sqrt{2}\)
• Dấu "=" xảy ra khi \(S_1=S_2=S_3\)
⇔ M là trọng tâm của ΔABC.