Cho bốn điểm A, B, C, D trong mặt phẳng. Chứng minh rằng \[\overrightarrow {AB} .\overrightarrow {CD} + \overrightarrow {BC} .\overrightarrow {AD} + \overrightarrow {CA} .\overrightarrow {BD} = 0.\]

Lời giải

Ta có:

• \(\overrightarrow {AD} = \overrightarrow {AB} + \overrightarrow {BC} + \overrightarrow {CD} \)

• \(\overrightarrow {CA} = \overrightarrow {BA} - \overrightarrow {BC} = - \overrightarrow {AB} - \overrightarrow {BC} \)

• \[\overrightarrow {BD} = \overrightarrow {BC} + \overrightarrow {CD} \]

Suy ra: \(\overrightarrow {AB} .\overrightarrow {CD} + \overrightarrow {BC} .\overrightarrow {AD} + \overrightarrow {CA} .\overrightarrow {BD} \)

\[ = \overrightarrow {AB} .\overrightarrow {CD} + \overrightarrow {BC} .\left( {\overrightarrow {AB} + \overrightarrow {BC} + \overrightarrow {CD} } \right) + \left( { - \overrightarrow {AB} - \overrightarrow {BC} } \right).\left( {\overrightarrow {BC} + \overrightarrow {CD} } \right)\]

\[ = \overrightarrow {AB} .\overrightarrow {CD} + \overrightarrow {BC} .\overrightarrow {AB} + {\overrightarrow {BC} ^2} + \overrightarrow {BC} .\overrightarrow {CD} - \overrightarrow {AB} .\overrightarrow {BC} - \overrightarrow {AB} .\overrightarrow {CD} - {\overrightarrow {BC} ^2} - \overrightarrow {BC} .\overrightarrow {CD} \]

= 0.

Vậy \[\overrightarrow {AB} .\overrightarrow {CD} + \overrightarrow {BC} .\overrightarrow {AD} + \overrightarrow {CA} .\overrightarrow {BD} = 0.\]