填七
設平面上有\(\Delta ABC\)與\(\Delta PQR\)，若\(2\vec{PA}+\vec{PB}+\vec{PC}=\vec{BC}\)，\(\vec{QA}+3\vec{QB}+\vec{QC}=\vec{CA}\)，\(\vec{RA}+\vec{RB}+4\vec{RC}=\vec{AB}\)，求\(\Delta PQR\)與\(\Delta ABC\)之面積比值 。
[解答]
可能太簡單，乏人問津，小弟自告奮勇。
\(2\overrightarrow{PA}+\overrightarrow{PB}+\overrightarrow{PC}=\overrightarrow{BC}\)
\(\overrightarrow{QA}+3\overrightarrow{QB}+\overrightarrow{QC}=\overrightarrow{CA}\)
\(\overrightarrow{RA}+\overrightarrow{RB}+4\overrightarrow{RC}=\overrightarrow{AB}\)
\(2\overrightarrow{PA}+\overrightarrow{PB}=\overrightarrow{BC}-\overrightarrow{PC}\)
\(3\overrightarrow{QB}+\overrightarrow{QC}=\overrightarrow{CA}-\overrightarrow{QA}\)
\(\overrightarrow{RA}+4\overrightarrow{RC}=\overrightarrow{AB}-\overrightarrow{RB}\)
\(2\overrightarrow{PA}+\overrightarrow{PB}=\overrightarrow{BP}\)
\(3\overrightarrow{QB}+\overrightarrow{QC}=\overrightarrow{CQ}\)
\(\overrightarrow{RA}+4\overrightarrow{RC}=\overrightarrow{AR}\)
\(\overrightarrow{PA}=\overrightarrow{BP}\)
\(3\overrightarrow{QB}=2\overrightarrow{CQ}\)
\(2\overrightarrow{RC}=\overrightarrow{AR}\)
....能找到P在\(\overline{AB}\)上，且\(\overline{AP}:\overline{PB}=1:1\)
Q在\(\overline{BC}\)上，\(\overline{BQ}:\overline{QC}=2:3\)
R在\(\overline{AC}\)上，\(\overline{AR}:\overline{RC}=2:1\)
方能找到\(\Delta PQR:\Delta ABC\)的比值了。
