Cho \[\Delta RSK\] và \[\Delta RSK\] có \(\frac{{RS}}{{PQ}} = \frac{{RK}}{{PM}} = \frac{{SK}}{{QM}}\), khi đó ta có

b) Ta có:  (cmt) suy ra \(\frac{{AD}}{{AF}} = \frac{{AB}}{{AC}}\) hay \(\frac{{AD}}{{AB}} = \frac{{AF}}{{AC}}.\)

Xét \[\Delta ABC\] và \[\Delta ADF\] có:

\(\widehat {BAC} = \widehat {DAF}\); \(\frac{{AD}}{{AB}} = \frac{{AF}}{{AC}}\;\left( {{\rm{cmt}}} \right)\)

Do đó $\Delta ABD\backsim \Delta ACF$ .

c) • Xét \[\Delta BEH\] và \[\Delta BDC\] có:

\(\widehat {EBH} = \widehat {DBC}\); \(\widehat {BEH} = \widehat {BDC}\;\left( { = 90^\circ } \right)\)

Do đó $\Delta BEH\backsim \Delta BDC\text{\hspace{0.17em}\hspace{0.17em}(g.g)}$ .

Suy ra \(\frac{{BE}}{{BD}} = \frac{{BH}}{{BC}}\) hay \(BH \cdot BD = BE \cdot BC\) (1)

• Xét \[\Delta CEH\] và \[\Delta CFB\] có:

\(\widehat {ECH} = \widehat {FCB}\); \(\widehat {CEH} = \widehat {CFB}\,\;\left( { = 90^\circ } \right)\).

Do đó $\Delta CEH\backsim \Delta CFB\text{\hspace{0.17em}\hspace{0.17em}(g.g)}$ .

Suy ra \(\frac{{CE}}{{CF}} = \frac{{CH}}{{CB}}\) hay \(CH \cdot CF = CE \cdot CB\)  (2)

Từ (1) và (2) ta có: \[BH \cdot BD + CH \cdot CF = BE \cdot BC + CE \cdot BC\]

\[ = BC\left( {BE + CE} \right) = BC \cdot BC = B{C^2}\] (đpcm).

• Mặt khác, ta có:

\(\frac{{HE}}{{AE}} + \frac{{HD}}{{BD}} + \frac{{HF}}{{CF}}\)\( = \frac{{\frac{1}{2} \cdot HE \cdot BC}}{{\frac{1}{2} \cdot AE \cdot BC}} + \frac{{\frac{1}{2} \cdot HD \cdot AC}}{{\frac{1}{2} \cdot BD \cdot AC}} + \frac{{\frac{1}{2} \cdot HF \cdot AB}}{{\frac{1}{2} \cdot CF \cdot AB}}\)

\( = \frac{{{S_{HBC}}}}{{{S_{ABC}}}} + \frac{{{S_{HAC}}}}{{{S_{BAC}}}} + \frac{{{S_{HAB}}}}{{{S_{CAB}}}}\)\( = \frac{{{S_{HBC}} + {S_{HAC}} + {S_{HAB}}}}{{{S_{ABC}}}} = \frac{{{S_{ABC}}}}{{{S_{ABC}}}} = 1\)