Dựng hbh BDFE
[tex]\Rightarrow \widehat{DFE}=\widehat{DBE}=\frac{1}{2}\widehat{ABC} \\ BD=E F, \ DF=BE[/tex]
Giả sử [tex]\widehat{ABC}\geq \widehat{ACB} \ (1)[/tex]
[tex]\Rightarrow AC\geq AB \\ \Leftrightarrow AC(AC+BC)\geq AB(AB+BC) \\ \Leftrightarrow \frac{AC}{AB+BC}\geq \frac{AB}{AC+BC}[/tex]
Lại có: [tex]\frac{BD}{BC}=\frac{AD}{AC}=\frac{AB}{AC+BC} \\ \frac{CE}{BC}=\frac{AE}{AB}=\frac{AC}{AB+BC}[/tex]
[tex]\Rightarrow \frac{CE}{BC}\geq \frac{BD}{BC}\Rightarrow CE\geq BD\Leftrightarrow CE\geq E F\Rightarrow \widehat{E FC}\geq \widehat{EC F}[/tex]
Xét ∆DFC cân tại D
[tex]\widehat{DFC}=\widehat{DCF} \\ \Rightarrow \widehat{DFE}+\widehat{EFC}=\widehat{DCE}+\widehat{EC F}[/tex]
[tex]\Rightarrow \widehat{DFE}\leq \widehat{DCE}[/tex]
[tex]\Rightarrow \widehat{ABC}\leq \widehat{ACB} \ (2)[/tex]
Từ (1)(2) => [tex]\widehat{ABC}=\widehat{ACB}[/tex] => đpcm
[TẶNG BẠN] TRỌN BỘ Bí kíp học tốt 08 môn
