\(\text{a)}\) Tam giác \(\text{ABC}\) cân tại \(\text{A}\) nên\(\text{ ABC = ACB}\) (t/c tam giác cân)
\(\Rightarrow\) \(\dfrac{\text{ABC}}{\text{2}}\) \(\text{=}\) \(\dfrac{\text{ACB}}{\text{2}}\)
Mà \(\text{ABD = CBD =}\) \(\dfrac{\text{ABC}}{\text{2}}\)
\(\text{ACE = BCE = }\dfrac{\text{ACB}}{\text{2}}\)
Nên \(\text{ABD = CBD = ACE = BCE}\)
Xét \(\Delta\text{EBC}\) và \(\Delta\text{DCB}\) có
\(\widehat{\text{EBC}}=\widehat{\text{DCB}}\text{(cmt)}\)
\(\text{BC}\) chung
\(\widehat{\text{ECB}}=\widehat{\text{DBC }}\text{(cmt)}\)
\(\Rightarrow\Delta\text{EBC}=\Delta\text{DCB}\text{(g.c.g)}\)
\(\text{⇒}\) \(\text{BE = CD}\) (\(\text{2}\) cạnh tương ứng)
Mà \(\text{AB = AC (gt)}\) nên \(\text{AB - BE = AC - CD}\)
\(\text{⇒}\) \(\text{AE = AD}\)
\(\text{⇒}\) \(\Delta\text{AED}\) cân tại \(\text{A}\) \(\text{(đpcm)}\)
\(\text{b)}\) \(\Delta\text{ABC}\) cân tại \(\text{A}\) \(\text{⇒}\) \(\widehat{\text{BAC}}\) \(\text{= 180}^{\text{o}}\) \(\text{- 2.ABC (1)}\)
\(\Delta\text{EAD}\) cân tại \(\text{A}\) \(\text{⇒}\) \(\widehat{\text{EAD}}\) \(\text{= 180}^{\text{o}}\)\(\text{- 2.AED (2)}\)
Từ \(\text{(1)}\) và \(\text{(2)}\) \(\text{⇒}\) góc \(\text{ABC = AED}\)
Mà \(\widehat{\text{ABC}}\) và \(\widehat{\text{AED}}\) là \(\text{2}\) góc ở vị trí đồng vị nên \(\text{ED // BC (đpcm)}\)
