Bài 11:

\(\begin{array}{l}a)\,\,\overrightarrow {MA} .\overrightarrow {MC}  = \overrightarrow {MB} .\overrightarrow {MD} \\VT = \,\overrightarrow {MA} .\overrightarrow {MC}  = \left( {\overrightarrow {MO}  + \overrightarrow {OA} } \right)\left( {\overrightarrow {MO}  + \overrightarrow {OC} } \right)\\ = {\overrightarrow {MO} ^2} + \overrightarrow {MO} .\overrightarrow {OC}  + \overrightarrow {MO} .\overrightarrow {OA}  + \overrightarrow {OA} .\overrightarrow {OC} \\ = M{O^2} + O{A^2} + \overrightarrow {MO} \left( {\overrightarrow {OC}  + \overrightarrow {OA} } \right)\\ = M{O^2} + O{A^2}.\\VP = \overrightarrow {MB} .\overrightarrow {MD}  = \left( {\overrightarrow {MO}  + \overrightarrow {OB} } \right)\left( {\overrightarrow {MO}  + \overrightarrow {OD} } \right)\\ = {\overrightarrow {MO} ^2} + \overrightarrow {MO} .\overrightarrow {OD}  + \overrightarrow {MO} .\overrightarrow {OC}  + \overrightarrow {OB} .\overrightarrow {OD} \\ = M{O^2} + O{B^2} + \overrightarrow {MO} \left( {\overrightarrow {OB}  + \overrightarrow {OD} } \right)\\ = M{O^2} + O{B^2}.\end{array}\)

Vì \(ABCD\) là hình chữ nhật \( \Rightarrow OA = OB\)

\(\begin{array}{l} \Rightarrow M{O^2} + O{A^2} = M{O^2} + O{B^2}\\ \Rightarrow VT = VP\\ \Leftrightarrow \overrightarrow {MA} .\overrightarrow {MC}  = \overrightarrow {MB} .\overrightarrow {MD} .\end{array}\)

\(\begin{array}{l}b)\,\,M{A^2} + \overrightarrow {MB} .\overrightarrow {MD}  = 2\overrightarrow {MA} .\overrightarrow {MO} \\ \Leftrightarrow M{A^2} + \overrightarrow {MA} .\overrightarrow {MC}  = 2\overrightarrow {MA} .\overrightarrow {MO} \,\,\,\left( {theo\,\,cm\,\,a)} \right)\\ \Leftrightarrow \overrightarrow {MA} \left( {\overrightarrow {MA}  + \overrightarrow {MC} } \right) = 2\overrightarrow {MA} .\overrightarrow {MO} \\ \Leftrightarrow \overrightarrow {MA} .2\overrightarrow {MO}  = 2\overrightarrow {MA} .\overrightarrow {MO} \,\,\,\,\left( {theo\,\,\,quy\,\,tac\,\,\,trung\,\,diem} \right)\end{array}\)