$AC^2=AF^2+CF^2=AD^2+DF^2+FC^2+2AD.DF$$=AD^2+DC^2+2AD.DF.$
$AB.AE+AD.AF=DC.(AB+BE)+BC(AD+DF)$
$=DC.AB+DC.BE+BC.AD+BC.DF=DC^2+AD^2+2AD.DF.$
(Vì $\triangle CFD\sim \triangle CEB$ $(g.g)\Rightarrow \frac{DC}{DF}=\frac{BC}{BE}\Rightarrow DC.BE=BC.DF$ )
Vậy $AC^2=AB.AE+AD.AF$.