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a) xét \(\Delta EAB\)và \(\Delta CAD\)có:

\(\hept{\begin{cases}AE=AC\left(gt\right)\\\widehat{EAB}=\widehat{DAC}\left(đđ\right)\\AB=AD\left(gt\right)\end{cases}}\)

\(\Rightarrow\Delta EAB=\Delta CAD\)(c - g - c)

\(\Rightarrow BE=DC\)( 2 cạnh tương ứng)

b) có \(\hept{\begin{cases}BE=2MB\left(gt\right)\\CD=2ND\left(gt\right)\\BE=CD\left(cmt\right)\end{cases}}\)

\(\Rightarrow MB=ND\)

\(\Delta EAB=\Delta CAD\left(cmt\right)\)

\(\Rightarrow\widehat{D}=\widehat{ABE}\)( 2 cạnh tương ứng )

xét \(\Delta DAN\)và\(\Delta BAM\)có

\(\hept{\begin{cases}ND=MB\left(cmt\right)\\\widehat{D}=\widehat{ABM}\left(cmt\right)\\AD=AB\left(gt\right)\end{cases}}\)

\(\Rightarrow\Delta DAN=\Delta BAM\left(c-g-c\right)\)

\(\Rightarrow\)AM = AN ( 2 cạnh tương ứng )

       \(\widehat{DAN}=\widehat{MAB}\)( 2 cạnh tương ứng )

mà \(\widehat{DAN}+\widehat{NAB}=180^o\left(kb\right)\)

\(\Rightarrow\widehat{MAB}+\widehat{NAB}=180^o\Rightarrow\widehat{MAN}=180^o\)

\(\Rightarrow\)M, N, A thẳng hàng

c) gọi BC cắt Ax tại P

\(\Rightarrow\hept{\begin{cases}BH\le BP\left(cgv\le ch\right)\\CK\le CP\left(cgv\le ch\right)\end{cases}}\)

\(\Rightarrow BH+CK\le BP+CP\)

\(\Rightarrow BH+CK\le BC\)

d) có\(BH+CK\le BC\left(cmt\right)\)

\(\Rightarrow GTLN\)của \(BH+CK=BC\)

dấu bằng xảy ra

\(\Leftrightarrow BH=BP;CK=CP\)

\(\Leftrightarrow H\equiv P;K\equiv P\)

\(\Leftrightarrow Ax\perp BC\)

\(\Rightarrow BH+CK\)lớn nhất