Ta có:
\(AD>AB-BD\) (BĐT trong \(\Delta ABD\) ) \(\left(1\right)\)
\(AD>AC-CD\) (BĐT trong \(\Delta ACD\) ) \(\left(2\right)\)
Từ \(\left(1\right)\left(2\right)\) cộng vế:
\(\Rightarrow2AD>AB-BD+AC-CD\\ \Rightarrow2AD>AB+AC-BC\\ \Rightarrow AD>\dfrac{AB+AC-BC}{2}\)
Tương tự, ta có:
\(AD< AB+BD\) (BĐT trong \(\Delta ABD\) ) \(\left(4\right)\)
\(AD< AC+CD\) (BĐT trong \(\Delta ACD\) ) \(\left(5\right)\)
Từ \(\left(4\right)\left(5\right)\), cộng vế:
\(\Rightarrow2AD< AB+BD+AC+CD\\ \Rightarrow2AD< AB+AC+BC\\ \Rightarrow AD< \dfrac{AB+AC+BC}{2}\)
mà
\(AD>\dfrac{AB+AC-BC}{2}\left(cmt\right)\\ \Rightarrow\dfrac{AB+AC-BC}{2}< AD< \dfrac{AB+AC+BC}{2}\)
\(AD>AB-BD\\ AD>AC-CD\\ \Rightarrow2.AD>AB+AC-BC\\ \Rightarrow AD>\dfrac{AB+AC-BC}{2}\)
\(AD< AB+BD\\ AD< AC+CD\\ \Rightarrow2.AD< AB+AC+BC\\ \Rightarrow AD< \dfrac{AB+AC+BC}{2}\)
