$$\eqalign{
& a)\,\,Ap\,\,dung\,\,HTL\,trong\,\,{\Delta _v}ABC: \cr
& A{B^2} = HB.BC \cr
& A{C^2} = HC.BC \cr
& \Rightarrow {{A{B^2}} \over {A{C^2}}} = {{HB} \over {HC}} = {\left( {{{AB} \over {AC}}} \right)^2} \cr
& b)\,\, \cr
& Ap\,\,dung\,\,HTL\,\,trong\,\,{\Delta _v}AHB:\,\,BM = {{B{H^2}} \over {AB}} \cr
& Ap\,\,dung\,\,HTL\,\,trong\,\,{\Delta _v}AHC:\,\,CN = {{C{H^2}} \over {AC}} \cr
& \Rightarrow {{BM} \over {CN}} = {{B{H^2}} \over {AB}}.{{AC} \over {C{H^2}}} = {\left( {{{BH} \over {CH}}} \right)^2}.{{AC} \over {AB}} = {\left( {{{AB} \over {AC}}} \right)^4}.{{AC} \over {AB}} = {\left( {{{AB} \over {AC}}} \right)^3} = {{A{B^3}} \over {A{C^3}}} \cr
& c)\,\,A{H^3} = BC.BM.CN \cr
& BC.BM.CN = BC.{{B{H^2}} \over {AB}}.{{C{H^2}} \over {AC}} \cr
& = {{BC.{{\left( {BH.CH} \right)}^2}} \over {AB.AC}} = {{BC.A{H^4}} \over {AH.BC}} = A{H^3} \cr} $$
