Ta có: \(\overrightarrow{IA}-2\cdot\overrightarrow{IB}+4\cdot\overrightarrow{IC}=\overrightarrow{0}\)
=>\(\overrightarrow{IA}-2\left(\overrightarrow{IA}+\overrightarrow{AB}\right)+4\left(\overrightarrow{IA}+\overrightarrow{AC}\right)=\overrightarrow{0}\)
=>\(3\cdot\overrightarrow{IA}-2\cdot\overrightarrow{AB}+4\cdot\overrightarrow{AC}=\overrightarrow{0}\)
=>\(3\cdot\overrightarrow{IA}=2\cdot\overrightarrow{AB}-4\cdot\overrightarrow{AC}\)
=>\(\overrightarrow{IA}=\frac23\cdot\overrightarrow{AB}-\frac43\cdot\overrightarrow{AC}\)
\(P=\overrightarrow{IA}\left(\overrightarrow{AB}+\overrightarrow{AC}\right)\)
\(=\left(\frac23\cdot\overrightarrow{AB}-\frac43\cdot\overrightarrow{AC}\right)\left(\overrightarrow{AB}+\overrightarrow{AC}\right)=\frac23\cdot\left(\overrightarrow{AB}\right)^2-\frac23\cdot\overrightarrow{AB}\cdot\overrightarrow{AC}-\frac43\cdot\left(\overrightarrow{AC}\right)^2\)
\(=\frac23\cdot AB^2-\frac23\cdot AB\cdot AC\cdot cosBAC-\frac43\cdot AC^2\)
\(=\frac23\cdot AB^2-\frac23\cdot AB^2\cdot cos60-\frac43\cdot AB^2=-\frac23\cdot AB^2-\frac23\cdot AB^2\cdot\frac12\)
\(=-AB^2=-a^2\)
