Using vector to solve four centers of triangle Note: generally, capital letters represent vectors, and vector * vector represents the product of two vectors 1. Prove that point O is the center of gravity of triangle ABC, which is AOB = BOC = COA 2. Prove: if h is a point in the plane of triangle ABC, and the square of the module of HA + the square of the module of BC = the square of the module of HB + the square of the module of Ca = the square of the module of HC + the square of the module of ab, then h is the perpendicular center of triangle ABC 3. Prove: if OA * (module of AB / AB - module of AC / AC) = ob * (module of Ba / BA - module of BC / BC) = OCT (module of Ca / Ca - module of CB / CB) = 0, then o is the inner core of triangle ABC 4. Prove that point O is a certain point on the plane, A.B.C is three non collinear points on the plane, the moving point P satisfies OP = OA + m (cosine value of the module and angle B of AB / AB + cosine value of the module and angle c of AC / AC), and M is greater than 0, then the trajectory of point P must pass the perpendicular center of triangle ABC

Using vector to solve four centers of triangle Note: generally, capital letters represent vectors, and vector * vector represents the product of two vectors 1. Prove that point O is the center of gravity of triangle ABC, which is AOB = BOC = COA 2. Prove: if h is a point in the plane of triangle ABC, and the square of the module of HA + the square of the module of BC = the square of the module of HB + the square of the module of Ca = the square of the module of HC + the square of the module of ab, then h is the perpendicular center of triangle ABC 3. Prove: if OA * (module of AB / AB - module of AC / AC) = ob * (module of Ba / BA - module of BC / BC) = OCT (module of Ca / Ca - module of CB / CB) = 0, then o is the inner core of triangle ABC 4. Prove that point O is a certain point on the plane, A.B.C is three non collinear points on the plane, the moving point P satisfies OP = OA + m (cosine value of the module and angle B of AB / AB + cosine value of the module and angle c of AC / AC), and M is greater than 0, then the trajectory of point P must pass the perpendicular center of triangle ABC

The center of gravity of a triangle is the intersection of its three midlines
Five center theorem of triangle
Theorem of center of gravity: the three middle lines of a triangle intersect at a point. The distance from this point to the vertex is twice the distance from it to the midpoint of the opposite side. This point is called the center of gravity of the triangle
Outer center theorem: the vertical bisectors of the three sides of a triangle meet at a point. This point is called the outer center of the triangle
Perpendicularity theorem: the three heights of a triangle intersect at a point. This point is called the perpendicularity of a triangle
Inner theorem: the bisectors of the three inner angles of a triangle meet at a point. This point is called the inner point of a triangle
Side center theorem: the bisector of an inner angle of a triangle intersects the bisector of the outer angle at the other two vertices at a point. This point is called the side center of a triangle. A triangle has three side centers
The center of gravity, outer center, perpendicular center, inner center and side center of a triangle are called the five centers of a triangle