Question

M thuộc BC: BD // AC; CE//AB

chứng minh \(\dfrac{1}{AM}=\dfrac{1}{AD}+\dfrac{1}{AE}\)

Answer 1

BD//AC nên theo hệ quả định lí talet:

\(\dfrac{AM}{MD}=\dfrac{MC}{MB}\Rightarrow\dfrac{AM}{AM+MD}=\dfrac{MC}{MC+MB}\Rightarrow\dfrac{AM}{AD}=\dfrac{MC}{BC}\)(1)

CE//AB nên theo hệ quả định lí talet:

\(\dfrac{AM}{ME}=\dfrac{BM}{MC}\Rightarrow\dfrac{AM}{AM+ME}=\dfrac{BM}{BM+MC}\Rightarrow\dfrac{AM}{AE}=\dfrac{BM}{BC}\left(2\right)\)

từ (1) và (2)

\(\Rightarrow\dfrac{AM}{AD}+\dfrac{AM}{AE}=\dfrac{MC}{BC}+\dfrac{BM}{BC}\\ \Leftrightarrow AM\left(\dfrac{1}{AD}+\dfrac{1}{AE}\right)=\dfrac{BC}{BC}=1\Rightarrow\dfrac{1}{AM}=\dfrac{1}{AE}+\dfrac{1}{AD}\)