Dựng $DJ \perp AB, EI \perp AC$
Ta có : $\begin{cases} DA^2=AJ. AB\\BD^2=BJ.BA\\AE^2=AI.AC\\CE^2=CI.CA\end{cases}$
$\Rightarrow \begin{cases} \dfrac{AD^2}{BD^2}=\dfrac{AJ}{BJ} \\\dfrac{CE^2}{AE^2}=\dfrac{CI}{AI}\end{cases}$
Lại có : $\triangle ABD$ ~ $\triangle ACE$
$\to \dfrac{AJ}{BJ} = \dfrac{CI}{AI} = \dfrac{BM}{CM}$
$\Rightarrow \begin{cases} MI//AB\\MJ//AC\end{cases}$
$\Rightarrow MIAJ$ là hình bình hành
$\to \begin{cases} MI=AJ\\MJ=AI\\ \widehat{MIC} =\widehat{MJB} =\widehat{IMJ} =\widehat{MIE} =\widehat{DJM} (1)\end{cases}$
+, $\triangle ABD$ ~ $\triangle CAE$
$\Rightarrow \triangle AIE$ ~ $\triangle DJA$
$\to \dfrac{EI}{AJ} = \dfrac{AI}{DJ}$
$\to \dfrac{EI}{MI} = \dfrac{MJ}{DJ} (2)$
$(1)(2) \Rightarrow \triangle EIM$ ~ $\triangle MJD$
$\to \widehat{IME} = \widehat{JDM}$
$\Leftrightarrow \widehat{IME} + \widehat{IMJ} + \widehat{JMD} = \widehat{JDM} + \widehat{MJB} + \widehat{DMJ}$
$\Rightarrow \widehat{EMD} = 90°$.
