• Chứng minh: \[\overrightarrow {AD} + \overrightarrow {BE} + \overrightarrow {CF} = \overrightarrow {AE} + \overrightarrow {BF} + \overrightarrow {CD} \]

\[\overrightarrow {AD} + \overrightarrow {BE} + \overrightarrow {CF} = \overrightarrow {AE} + \overrightarrow {ED} + \overrightarrow {BF} + \overrightarrow {FE} + \overrightarrow {CD} + \overrightarrow {DF} \]

\[ = \left( {\overrightarrow {AE} + \overrightarrow {BF} + \overrightarrow {CD} } \right) + \left( {\overrightarrow {ED} + \overrightarrow {DF} + \overrightarrow {FE} } \right)\]

\[ = \overrightarrow {AE} + \overrightarrow {BF} + \overrightarrow {CD} + \overrightarrow {EF} + \overrightarrow {FE} \]

\[\overrightarrow {AE} + \overrightarrow {BF} + \overrightarrow {CD} \] (đpcm) (1)

• Chứng minh: \[\overrightarrow {AE} + \overrightarrow {BF} + \overrightarrow {CD} = \overrightarrow {AF} + \overrightarrow {BD} + \overrightarrow {CE} \]

\[\overrightarrow {AE} + \overrightarrow {BF} + \overrightarrow {CD} \]

\[ = \overrightarrow {AF} + \overrightarrow {FE} + \overrightarrow {BD} + \overrightarrow {DF} + \overrightarrow {CE} + \overrightarrow {ED} \]

\[ = \overrightarrow {AF} + \overrightarrow {BD} + \overrightarrow {CE} + \left( {\overrightarrow {FE} + \overrightarrow {ED} + \overrightarrow {DF} } \right)\]

\[ = \overrightarrow {AF} + \overrightarrow {BD} + \overrightarrow {CE} \](đpcm) (2)

Từ (1) và (2): \[\overrightarrow {AB} + \overrightarrow {BE} + \overrightarrow {CF} = \overrightarrow {AE} + \overrightarrow {BF} + \overrightarrow {CD} = \overrightarrow {AF} + \overrightarrow {BD} + \overrightarrow {CE} \].