*Tỉ lệ diện tích \(\dfrac{S_{DEF}}{S_{ABC}}\) chứ nhỉ?
- Vì \(\dfrac{EA}{EB}=3;EA+EB=AB\)
\(\Rightarrow\left\{{}\begin{matrix}\dfrac{EA}{EA+EB}=\dfrac{3}{1+3}\\\dfrac{EB}{EA+EB}=\dfrac{1}{1+3}\end{matrix}\right.\) \(\Rightarrow\left\{{}\begin{matrix}\dfrac{EA}{AB}=\dfrac{3}{4}\\\dfrac{EB}{AB}=\dfrac{1}{4}\end{matrix}\right.\)
- Tương tự: \(\left\{{}\begin{matrix}\dfrac{DB}{BC}=\dfrac{3}{4}\\\dfrac{DC}{BC}=\dfrac{1}{4}\end{matrix}\right.\) ; \(\Rightarrow\left\{{}\begin{matrix}\dfrac{FC}{AC}=\dfrac{3}{4}\\\dfrac{FA}{AC}=\dfrac{1}{4}\end{matrix}\right.\)
- Ta có: \(\dfrac{S_{AEF}}{S_{AEC}}=\dfrac{AF}{AC}\) (cùng chung đường cao đỉnh A).
\(\dfrac{S_{AEC}}{S_{ABC}}=\dfrac{AE}{AB}\) (cùng chung đường cao đỉnh C).
\(\Rightarrow\dfrac{S_{AEF}}{S_{AEC}}.\dfrac{S_{AEC}}{S_{ABC}}=\dfrac{AF}{AC}.\dfrac{AE}{AB}\)
\(\Rightarrow\dfrac{S_{AEF}}{S_{ABC}}=\dfrac{AF}{AC}.\dfrac{AE}{AB}=\dfrac{1}{4}.\dfrac{3}{4}=\dfrac{3}{16}\left(1\right)\)
- Tương tự: \(\left\{{}\begin{matrix}\dfrac{S_{BDE}}{S_{ABC}}=\dfrac{DB}{BC}.\dfrac{BE}{AB}=\dfrac{3}{4}.\dfrac{1}{4}=\dfrac{3}{16}\left(2\right)\\\dfrac{S_{DFC}}{S_{ABC}}=\dfrac{DC}{AB}.\dfrac{FC}{AC}=\dfrac{1}{4}.\dfrac{3}{4}=\dfrac{3}{16}\left(3\right)\end{matrix}\right.\)
- Lấy (1) + (2) + (3), ta có:
\(\dfrac{S_{AEF}}{S_{ABC}}+\dfrac{S_{BDE}}{S_{ABC}}+\dfrac{S_{DFC}}{S_{ABC}}=\dfrac{9}{16}\)
\(\Rightarrow\dfrac{S_{ABC}-S_{DEF}}{S_{ABC}}=\dfrac{9}{16}\)
\(\Rightarrow1-\dfrac{S_{DEF}}{S_{ABC}}=\dfrac{9}{16}\)
\(\Rightarrow\dfrac{S_{DEF}}{S_{ABC}}=\dfrac{7}{16}\)
