`a)`  Xét `ΔBCE` và `ΔCDF` có:

`CF=BE`

`CD=CB`

`hat{CBE}=hat{DCF}=90^o`

`→` `ΔBCE=ΔCDF` `(c-g-c)`

`→` `hat{CFM}=hat{CEB}`

Xét `ΔCFM` và `ΔCEB` có:

`hat{CFM}=hat{CEB}`

`hat{BCE}` chung

`→` `ΔCFM` $\backsim$ `ΔCEB` `(g-g)`

`→` `hat{CMF}=hat{CBE}=90^o`

`→` `CE⊥DF`

`b)` Vì `ΔCFM` $\backsim$ `ΔCEB` `(g-g)`

`→` `(CF)/(CE)=(CM)/(CB)`

`→` `CB.CF=CM.CE`

`→` `CB=(CM.CE)/(CF)`

`→` `a=(CM.CE)/(CF)`

`c)` Xét `ΔCMD` và `ΔDCF` có:

`hat{CMD}=hat{DCE}=90^o`

`hat{CDM}` chung

`→` `ΔCDM` $\backsim$ `ΔDCF` `(g-g)`

`→` `(CM)/(CD)=(CF)/(DF)`

`→` `CM=(CF.CD)/(DF)` 

Ta có: `CF.CD=a. 1/2 a=1/2 a^2`

`→` `CM=(1/2 a^2)/(DF)`

Vì `ΔCDM` $\backsim$ `ΔDCF` `(g-g)` 

`→` `(DM)/(DC)=(DC)/(DF)`

`→` `DM=(DC.DC)/(DF)`

hay `DM=a^2/(DF)`

`→` `DM.CM=(1/2 a^4)/(DF^2)`

Ta có: `DF^2=DC^2+CF^2`

`→` `DF^2=a^2+1/4a^2`

`→` `DF^2=5/4 a^2`

`→` `DM.CM=(1/2a^4)/(5/4a^2)=2/5 a^2`

`→` `S_{ΔMDC}=1/2 DM.CM=1/2 . 2/5.a^2=1/5a^2`