\(\left\{{}\begin{matrix}\widehat{AHD}=\widehat{BHI}=90^o\\\widehat{ADH}=\widehat{IBH}\left(phu.\widehat{HDK}\right)\end{matrix}\right.\)\(\Rightarrow\Delta ADH\sim\Delta IBH\left(g-g\right)\)
\(\Rightarrow\frac{AD}{AH}=\frac{IB}{IH}\)
\(\Rightarrow\frac{BC}{AH}=\frac{BI}{IH}\left(hcn\right)\Rightarrow\frac{BI}{BC}=\frac{IH}{AH}\)
Mà CD // AH nên theo Ta-lét ta có:
\(\frac{BI}{BC}=\frac{IA}{AK}\) \(\Rightarrow\frac{AI}{AK}=\frac{IH}{AH}\)
\(\Rightarrow AI\cdot AH=AK\cdot IH\)
\(\Leftrightarrow AI\cdot AH+AK\cdot AH=AK\cdot AH+AK\cdot IH\) (cộng 2 vế)
\(\Leftrightarrow AI\cdot AH+AK\cdot AH=AK\left(HA+IH\right)\)
\(\Leftrightarrow AH\left(AI+AK\right)=AK\cdot AI\)
\(\Leftrightarrow\frac{1}{AH\left(AI+AK\right)}=\frac{1}{AK\cdot AI}\)
\(\Leftrightarrow\frac{AI+AK}{AH\left(AI+AK\right)}=\frac{AI+AK}{AI\cdot AK}\)
\(\Leftrightarrow\frac{1}{AH}=\frac{1}{AK}+\frac{1}{AI}\) (đpcm)
