Bài 20 trang 18 Sách giáo khoa (SGK) Hình học 10 Nâng cao
Cho sáu điểm A, B, C, D, E, F. Chứng minh rằng ...
Cho sáu điểm A, B, C, D, E, F. Chứng minh rằng
Bài 20. Cho sáu 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} ).
Hướng dẫn trả lời
Theo quy tắc ba điểm, ta có
(eqalign{
& overrightarrow {AD} + overrightarrow {BE} + overrightarrow {CF} = left( {overrightarrow {AE} + overrightarrow {ED} }
ight) + left( {overrightarrow {BF} + overrightarrow {FE} }
ight) + left( {overrightarrow {CD} + overrightarrow {DF} }
ight) cr
& ,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,, = overrightarrow {AE} + overrightarrow {BF} + overrightarrow {CD} + left( {overrightarrow {FE} + overrightarrow {ED} + overrightarrow {DF} }
ight) cr
& ,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,, = overrightarrow {AE} + overrightarrow {BF} + overrightarrow {CD} + left( {overrightarrow {FD} + overrightarrow {DF} }
ight) cr
& ,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,, = overrightarrow {AE} + overrightarrow {BF} + overrightarrow {CD} cr} )
Tương tự, ta cũng có
(eqalign{
& overrightarrow {AD} + overrightarrow {BE} + overrightarrow {CF} = left( {overrightarrow {AF} + overrightarrow {FD} }
ight) + left( {overrightarrow {BD} + overrightarrow {DE} }
ight) + left( {overrightarrow {CE} + overrightarrow {EF} }
ight) cr
& ,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,, = overrightarrow {AF} + overrightarrow {BD} + overrightarrow {CE} + left( {overrightarrow {FD} + overrightarrow {DE} + overrightarrow {EF} }
ight) cr
& ,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,, = overrightarrow {AF} + overrightarrow {BD} + overrightarrow {CE} + left( {overrightarrow {FE} + overrightarrow {EF} }
ight) cr
& ,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,, = overrightarrow {AF} + overrightarrow {BD} + overrightarrow {CE} cr} )
Vậy ta có (overrightarrow {AD} + overrightarrow {BE} + overrightarrow {CF} = overrightarrow {AE} + overrightarrow {BF} + overrightarrow {CD} = overrightarrow {AF} + overrightarrow {BD} + overrightarrow {CE} )
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