Cho tam giác đều ABC có O là trọng tâm và M là một điểm tùy ý trong tam giác. Gọi D, E, F lần lượt là chân đường vuông góc hạ từ M đền BC, AC, AB. Chứng minh rằng :
\(\overrightarrow{MD}+\overrightarrow{ME}+\overrightarrow{MF}=\dfrac{3}{2}\overrightarrow{MO}\)
Qua M kẻ các đường thẳng song song với các cạnh của tam giác
A1B1 // AB; A2C2 // AC; B2C1 // BC.
Dễ thấy các tam giác MB1C2; MA1C1;MA2B2 đều là các tam giác đều. Ta lại có MD
B1C2 nên MD cũng là trung điểm thuộc cạnh B1C2 của tam giác MB1C2
Ta có 2
=
+ ![This is the rendered form of the equation. You can not edit this directly. Right click will give you the option to save the image, and in most browsers you can drag the image onto your desktop or another program.](http://img.loigiaihay.com/picture/article/2017/0214/bai-9-trang-17-sgk-hinh-hoc-lop-10_10_1487087528.jpg)
Tương tự: 2
=
+ ![This is the rendered form of the equation. You can not edit this directly. Right click will give you the option to save the image, and in most browsers you can drag the image onto your desktop or another program.](http://img.loigiaihay.com/picture/article/2017/0214/bai-9-trang-17-sgk-hinh-hoc-lop-10_13_1487087528.jpg)
2
=
+![This is the rendered form of the equation. You can not edit this directly. Right click will give you the option to save the image, and in most browsers you can drag the image onto your desktop or another program.](http://img.loigiaihay.com/picture/article/2017/0214/bai-9-trang-17-sgk-hinh-hoc-lop-10_16_1487087528.jpg)
=> 2(
+
+
) = (
+
) + (
+
) + (
+
)
Tứ giác là hình bình hành nên
Tương tự:
+
= ![This is the rendered form of the equation. You can not edit this directly. Right click will give you the option to save the image, and in most browsers you can drag the image onto your desktop or another program.](http://img.loigiaihay.com/picture/article/2017/0214/bai-9-trang-17-sgk-hinh-hoc-lop-10_31_1487087528.jpg)
=> 2(
+
+
) =
+
+![This is the rendered form of the equation. You can not edit this directly. Right click will give you the option to save the image, and in most browsers you can drag the image onto your desktop or another program.](http://img.loigiaihay.com/picture/article/2017/0214/bai-9-trang-17-sgk-hinh-hoc-lop-10_34_1487087528.jpg)
vì O là trọng tâm bất kì của tam giác và M là một điểm bất kì nên
Cuối cùng ta có:
2(
+
+
) = 3
;
=>
+
+
= ![This is the rendered form of the equation. You can not edit this directly. Right click will give you the option to save the image, and in most browsers you can drag the image onto your desktop or another program.](http://img.loigiaihay.com/picture/article/2017/0214/bai-9-trang-17-sgk-hinh-hoc-lop-10_4_1487087528.jpg)
![This is the rendered form of the equation. You can not edit this directly. Right click will give you the option to save the image, and in most browsers you can drag the image onto your desktop or another program.](http://img.loigiaihay.com/picture/article/2017/0214/bai-9-trang-17-sgk-hinh-hoc-lop-10_5_1487087528.jpg)