Five-center Theorem of Triangle The center of gravity, outer center, vertical center and inner center of a triangle are called the four centers of a triangle. The four-center theorem of triangle refers to the triangle's center of gravity theorem, outer center theorem, vertical center theorem and inner center theorem. First, the triangle center of gravity theorem The median lines of three sides of a triangle intersect at one point. This point is called the center of gravity of the triangle. (The center of gravity was originally a physical concept. For a triangular thin plate with the same thickness and uniform mass, its center of gravity is just the intersection of the three midlines of the triangle, hence the name. The nature of the center of gravity is 1, and the ratio of the distance from the center of gravity to the vertex to the distance from the center of gravity to the midpoint of the opposite side is 2 1. 2. The areas of the three triangles formed by the center of gravity and the three vertices of the triangle are equal. That is, the distance from the center of gravity to the three sides is inversely proportional to the growth of the three sides. 3. The sum of squares of the distances from the center of gravity to the three vertices of the triangle is the smallest. 4. In the plane rectangular coordinate system, the coordinate of the center of gravity is the arithmetic average of the vertex coordinates, that is, the coordinate of the center of gravity is ((X 1+X2+X3)/3, (Y 1+Y2+Y3)/3. Second, the triangle center theorem The center of the circumscribed circle of a triangle is called the center of the triangle. The property of the epicentre is 1, and the perpendicular lines of the three sides of the triangle intersect at a point, which is the epicentre of the triangle. 2. If O is the outer center of △ABC, ∠BOC=2∠A(∠A is acute angle or right angle) or ∠ BOC = 360-2 ∠ A (∠ A is obtuse angle). 3. When the triangle is an acute triangle, the outer center is inside the triangle; When the triangle is an obtuse triangle, the outer center is outside the triangle; When the triangle is a right triangle, the outer center is on the hypotenuse and coincides with the midpoint of the hypotenuse. 4. To calculate the coordinates of the epicenter, we must first calculate the following temporary variables: d 1, d2 and d3 are the point multiplication of the vectors whose three vertices are connected with the other two vertices. c 1=d2d3，c2=d 1d3，C3 = d 1 D2； C=c 1+c2+c3. Coordinate of gravity center: ((c2+c3)/2c, (c 1+c3)/2c, (c 1+c2)/2c). 5. The distances from the outer center to the three vertices are equal. 3. The theorem of the vertical center of a triangle The three heights (straight lines) of a triangle intersect at a point, which is called the vertical center of a triangle. The nature of the vertical center: 1, three vertices and three vertical feet of a triangle. You can get six four-point circles by hanging these seven points. 2. Triangular line with three points * * * o, g, h, OG \u GH = 1 \u 2. (This straight line is called the Euler line of triangle) 3. The distance from the vertical center to the vertex of the triangle is twice as long as the distance from the outer center of the triangle to the opposite side of the vertex. The product of two parts of each high line is equal. Theorem proof shows that in Δ δABC, AD and BE are two heights, intersecting at point O, connecting CO and extending the intersection point AB to point F, proof: CF⊥AB proof: connecting de≈ADB =∠aeb = 90 degrees ∴A, B, D and E * * * circle ∠ Ade. Fourth, the triangle center theorem The center of the inscribed circle of a triangle is called the heart of the triangle. Intrinsic property: 1, the three bisectors of a triangle intersect at one point. This point is the center of the triangle. 2. The distance from the center to the right-angled triangle edge is equal to the sum of the two right-angled edges minus half the difference of the hypotenuse. 3.p is any point on the Δ ABC plane, and the necessary and sufficient condition for point I to be a Δ ABC kernel is: vector PI=(a× vector PA+b× vector PB+c× vector PC)/(a+b+c). 4.o is the core of the triangle, and A, B and C are the three vertices of the triangle. If the intersection of AO and BC is longer than n, there is AO.