Đáp án:
$\begin{array}{l}
a)Theo\,Pytago:\\
D{F^2} + D{E^2} = F{E^2}\\
\Leftrightarrow D{F^2} = {8^2} - {6^2} = 64 - 36 = 28\\
\Leftrightarrow DF = 2\sqrt 7 \left( {cm} \right)\\
Theo\,t/c:\\
\dfrac{{ID}}{{DE}} = \dfrac{{IF}}{{EF}} = \dfrac{{ID}}{6} = \dfrac{{IF}}{8} = \dfrac{{ID + IF}}{{6 + 8}}\\
= \dfrac{{2\sqrt 7 }}{{14}} = \dfrac{{\sqrt 7 }}{7}\\
\Leftrightarrow \left\{ \begin{array}{l}
ID = \dfrac{{6\sqrt 7 }}{7}\left( {cm} \right)\\
IF = \dfrac{{8\sqrt 7 }}{7}\left( {cm} \right)
\end{array} \right.\\
b)Xet:\Delta DEF;\Delta KIF:\\
+ \widehat {EDF} = \widehat {IKF} = {90^0}\\
+ \widehat F\,chung\\
\Leftrightarrow \Delta DEF \sim \Delta KIF\left( {g - g} \right)\\
c)Do:\Delta DEF \sim \Delta KIF\\
\Leftrightarrow \dfrac{{KI}}{{DE}} = \dfrac{{IF}}{{EF}}\\
\Leftrightarrow KI.EF = DE.IF
\end{array}$
