Cho 6 điểm A, B, C, D, E, F. Chứng minh rằng:

\(\overrightarrow {AD} + \overrightarrow {BE} + \overrightarrow {CF} = \overrightarrow {AE} + \overrightarrow {BF} + \overrightarrow {CD} = \overrightarrow {AF} + \overrightarrow {BD} + \overrightarrow {CE} \)

Ta có:

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

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

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

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

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

\[ = \overrightarrow {AE} + \overrightarrow {BF} + \overrightarrow {CD} + \overrightarrow 0 \]

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

Suy ra \(\overrightarrow {AD} + \overrightarrow {BE} + \overrightarrow {CF} = \overrightarrow {AE} + \overrightarrow {BF} + \overrightarrow {CD} \)

Chứng minh tương tự ta cũng có: \(\overrightarrow {AD} + \overrightarrow {BE} + \overrightarrow {CF} = \overrightarrow {AF} + \overrightarrow {BD} + \overrightarrow {CE} \)

Vậy \(\overrightarrow {AD} + \overrightarrow {BE} + \overrightarrow {CF} = \overrightarrow {AE} + \overrightarrow {BF} + \overrightarrow {CD} = \overrightarrow {AF} + \overrightarrow {BD} + \overrightarrow {CE} \)