$a)$Xét $\Delta MAE$ và $\Delta MCA$
$\widehat{M_1}:$ chung
$\widehat{A_1}=\widehat{C_1}($ Chắn hai cũng bằng nhau $AM,MB)$
$\Rightarrow \Delta MAE \backsim \Delta MCA\\ b)\Delta MAE \backsim \Delta MCA\\ \Rightarrow \dfrac{ME}{MA}=\dfrac{MA}{MC}\\ \Leftrightarrow ME.MC=MA^2(1)$
Tương tự $\Delta MBF \backsim \Delta MDB$
$\Rightarrow \dfrac{MF}{MB}=\dfrac{MB}{MD}\\ \Leftrightarrow MF.MD=MB^2(2)$
$(1)(2)$ mà $MA=MB$(Do $M$ nằm chính giữa cung $AB)$
$\Rightarrow ME.MC = MF.MD$
$c)ABCD$ nội tiếp
$\Rightarrow \widehat{CAE}+\widehat{CDB}=180^o$
$\widehat{C_1}=\widehat{D_2}$(Chắn hai cũng bằng nhau $AM,MB)$
$\Delta AEC, \widehat{E_1}=\widehat{C_1}+\widehat{CAE}\\ =\widehat{C_1}+180^o-\widehat{CDB}\\ =\widehat{C_1}+180^o-\widehat{D_1}-\widehat{D_2}\\ =180^o-\widehat{D_1}\\ \Leftrightarrow \widehat{E_1}+\widehat{D_1}=180^o$
$\Rightarrow CEFD$ nội tiếp được
$d)OA=OM=AM=R$
$\Rightarrow \Delta OAM$ đều.
