In the college entrance examination, "vector" is often used as a carrier to investigate the "four hearts" of a triangle. Their vector expressions have many important properties, which always lead to some novel and unique questions, which not only examine knowledge points such as vectors, but also cultivate candidates' ability to analyze and solve problems. This requires us to understand the geometric meaning of vectors on the basis of being familiar with the "four centers" theorem of triangles and the algebraic operation of vectors.
The four centers of a triangle
1, the perpendicular lines of the three sides of the triangle intersect at a point, which is the center of the circumscribed circle of the triangle.
2. There is only one circumscribed circle of a triangle, that is, for a given triangle, its outer center is unique, but there are countless inscribed triangles of a circle, and the outer centers of these triangles coincide.
3. The outer center of the acute triangle is in the triangle; The outer center of an obtuse triangle is outside the triangle; The outer center of a right triangle coincides with the midpoint of the hypotenuse.
4，OA=OB=OC=R .
5，∠BOC=2∠BAC，∠AOB=2∠ACB，∠COA=2∠CBA .
6，S△ABC=abc/4R .