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 trang 18 Sách giáo khoa (SGK) Hình học 10 Nâng cao – Bài 3. Hiệu của hai vectơ Bài 20 . Cho sáu điểm (A, B, C, D, E, F). Chứng minh rằng (overrightarrow {AD} + overrightarrow {BE} + overrightarrow {CF} = overrightarrow {AE} + ...
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} )
