Cho các điểm A, B, C, D, E, F. Chứng minh rằng:
\[\overrightarrow {AD} + \overrightarrow {BE} + \overrightarrow {CF} = \overrightarrow {AE} + \overrightarrow {BF} + \overrightarrow {CD} \].
VT = \[\overrightarrow {AD} + \overrightarrow {BE} + \overrightarrow {CF} \]
\[ = \overrightarrow {AE} + \overrightarrow {ED} + \overrightarrow {BF} + \overrightarrow {FE} + \overrightarrow {CD} + \overrightarrow {DF} \]
\[ = \overrightarrow {AE} + \overrightarrow {BF} + \overrightarrow {CD} + \left( {\overrightarrow {ED} + \overrightarrow {DF} } \right) + \overrightarrow {FE} \]
\[ = \overrightarrow {AE} + \overrightarrow {BF} + \overrightarrow {CD} + \left( {\overrightarrow {EF} + \overrightarrow {FE} } \right)\]
\[ = \overrightarrow {AE} + \overrightarrow {BF} + \overrightarrow {CD} + \overrightarrow {EE} \]
\[ = \overrightarrow {AE} + \overrightarrow {BF} + \overrightarrow {CD} \](do \[\overrightarrow {EE} = \overrightarrow 0 \]) = VP
Vậy \[\overrightarrow {AD} + \overrightarrow {BE} + \overrightarrow {CF} = \overrightarrow {AE} + \overrightarrow {BF} + \overrightarrow {CD} \].