\(a,Xét\Delta AHBvà\Delta AHMcó\)
\(AB=AM\left(gt\right)\)
\(\widehat{A1}=\widehat{A2}\left(AHlàtiaphângiáccủa\widehat{A}\right)\)
\(AHlàcạnhchung\)
\(\Rightarrow\Delta AHB=\Delta AHM\left(c-g-c\right)\left(đpcm\right)\)
\(b,Tacó\widehat{ABH}+\widehat{HBD}=180^0\left(k/bù\right)\)
\(Và:\widehat{AMH}+\widehat{HMC}=180^0\left(kề/bù\right)\)
\(Mà:\widehat{ABH}=\widehat{AMH}\left(\Delta ABH=\Delta AMH\right)\)
\(\Rightarrow\widehat{HBD}=\widehat{HMC}\)
\(Xét\Delta BHDvà\Delta MHCcó:\)
\(BH=MH\left(\Delta AHB=\Delta AHM\right)\)
\(\widehat{BHD}=\widehat{MHC}\left(đ/đỉnh\right)\)
\(\widehat{HBD}=\widehat{HMC}\left(cmt\right)\)
\(\Rightarrow\Delta BHD=\Delta MHC\left(g-c-c\right)\)
\(\Rightarrow HD=HC\left(2c.t.ứ\right)\)
Lại có: \(\left\{{}\begin{matrix}BC=BH+HC\\MD=MH+HD\end{matrix}\right.\)
Mà: \(\left\{{}\begin{matrix}BH=MH\left(cmt\right)\\HC=HD\left(cmt\right)\end{matrix}\right.\)
\(MD=BC\left(đpcm\right)\)
\(c,Chứngminhtươngtựtađược:AD=AC\)
\(Xét\Delta ADHvà\Delta ACHcó:\)
\(\widehat{A1}=\widehat{A2}\)
\(AD=AC\left(cmt\right)\)
\(AHlàcạnhchung\)
\(\Rightarrow\Delta ADH=\Delta ACH\left(c-g-c\right)\)
\(\Rightarrow\widehat{AHD}=\widehat{AHC}\left(2.g.t.ứ\right)\)
\(Mà:\widehat{AHD}+\widehat{AHC}=180^0\)
\(\Rightarrow\widehat{AHD}=\widehat{AHC}=\frac{180^0}{2}=90^0\)
\(\Rightarrow AH\perp CD\)