1. Xét ΔAIE và ΔAIF có:
Chung AI
\(\widehat{EAI}=\widehat{FAI}\)(gt)
\(AE=AF\)(gt)
⇒ΔAIE = ΔAIF (c.g.c)
2. Xét ΔABC có: \(\widehat{BAC}+\widehat{B}+\widehat{BCA}=180^o\\ \Rightarrow\widehat{BAC}+\widehat{BCA}=120^o\\ \Rightarrow2\widehat{ICA}+2\widehat{IAC}=120^o\\ \Rightarrow\widehat{ICA}+\widehat{IAC}=60^o\)
Xét ΔAIC có: \(\widehat{ICA}+\widehat{IAC}+\widehat{AIC}=180^o\\ \Rightarrow\widehat{AIC}=120^o\)
\(\Rightarrow\widehat{EID}=\widehat{AIC}=120^o\) (2 góc đối đỉnh)
Ta có: \(\widehat{AIE}+\widehat{EID}=180^o\\ \Rightarrow\widehat{AIE}=60^o\)
3. ΔAIE = ΔAIF(cma)
\(\Rightarrow\widehat{AIE}=\widehat{AIF}=60^o\) (2 góc tương ứng)
\(\widehat{AIE}=\widehat{CIF}=60^o\) (2 góc đối đỉnh)
Xét ΔCID và ΔCIF có:
\(\widehat{CID}=\widehat{CIF}\left(=60^o\right)\)
Chung CI
\(\widehat{ICD}=\widehat{ICF}\)(gt)
⇒ΔCID = ΔCIF(g.c.g)
⇒ID=IF (2 cạnh tương ứng)
Mà \(IE=IF\)(cmb) \(\Rightarrow IE=ID\)
