$$\eqalign{
& a)\,\,A{B^2} + B{C^2} = {6^2} + {8^2} = 100 = A{C^2} \cr
& \Rightarrow \Delta ABC\,\,vuong\,\,tai\,\,B \cr
& b)\,\,Xet\,\,tu\,\,giac\,\,BEHF\,co: \cr
& \widehat {EBF} = \widehat {BEH} = \widehat {BFH} = {90^0} \cr
& \Rightarrow BEGF\,\,la\,\,HCN\,\,\left( {dhnb} \right) \cr
& c)\,\,Ap\,\,dung\,\,HTL\,\,trong\,\,tam\,\,giac\,\,vuong\,\,ABH: \cr
& B{H^2} = BE.BA \cr
& Ap\,\,dung\,\,HTL\,\,trong\,\,tam\,\,giac\,\,vuong\,\,BCH: \cr
& B{H^2} = BF.BC \cr
& \Rightarrow BE.BA = BF.BC \cr
& d)\,\,Ap\,\,dung\,\,HTL\,\,trong\,\,\Delta ABC: \cr
& BH = {{BA.BC} \over {AC}} = {{6.8} \over {10}} = 4,8\,\,\left( {cm} \right) \cr
& B{H^2} = BE.BA \Rightarrow BE = {{B{H^2}} \over {BA}} = {{4,{8^2}} \over 6} = 3,84\,\,\left( {cm} \right) = HF \cr
& TT:\,\,BF = {{B{H^2}} \over {BC}} = 2,88\,\,\left( {cm} \right) = HE \cr
& \Rightarrow {S_{HEF}} = {1 \over 2}HE.HF = {1 \over 2}.3,84.2,88 = 5,5296\,\,\left( {c{m^2}} \right) \cr} $$
