Five Centers of Triangle and Their Properties

Five centers of triangle and their properties;

Five hearts: center of gravity, outer heart, hanging heart, inner heart and lateral heart.

Nature of the center of gravity:

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.

The nature of the external world:

1. The perpendicular bisector of three sides of a triangle intersect at a point, which is the outer center 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.

The essence of the heart:

1, three vertices and three vertical feet of a triangle, and these seven points can get six four-point circles.

2. The triangle three-point * * line of the outer center O, the center of gravity G and the vertical center H, OG∶GH= 1∶2.

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.

Intrinsic essence:

1, the three bisectors of the 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 plane of Δ ABC, and the necessary and sufficient conditions for point I to be the heart of Δ ABC are: vector PI=(a× vector PA+b× vector PB+c× vector PC)/(a+b+c).

4.o is the heart of the triangle, and A, B and C are the three vertices of the triangle. If the intersection of AO and BC extends to n, there is AO: On = AB: BN = AC: CN = (AB+AC): BC.

Nature of lateral center:

1, the bisector of an inner corner of a triangle and the bisector of an outer corner of the other two vertices intersect at a point, which is the edge center of the triangle.

2. Every triangle has three side centers.

3. The distance from the side center to the three sides is equal.