The epicenter is the intersection of three sides of perpendicular bisector, and its distances to the three vertices are equal.

The center of gravity is the intersection of three midlines, and its distance from the vertex is twice as long as its distance from the midpoint of the opposite side.

The vertical center is the intersection of three heights, which can form many right triangles.

Paracenter is the intersection of an inner bisector and its two non-adjacent outer bisectors, and its distances to three sides are equal. (1) The areas of the three triangles formed by connecting the center of gravity with the three vertices are equal;

(2) The distances from the outer center to the three vertices are equal;

(3) Among the four points, the vertical center and the three vertices, any point is the vertical center of the triangle formed by the other three points;

(4) The distances from the medial center and the lateral center to the three sides are equal;

(5) The vertical center is the heart of a triangle composed of three vertical feet;

(6) The outer center is the vertical center of the midpoint triangle;

(7) The center is also the center of gravity of the midpoint triangle;

(8) The center of a triangle is also the center of its vertical triangle. Five-center theorem of triangle

Gravity center theorem: the three midlines of a triangle intersect at a point, and the distance from the point to the vertex is twice as long as the distance from the midpoint of the opposite side to the vertex, which is called the gravity center of the triangle.

Eccentricity theorem: the perpendicular lines of three sides of a triangle intersect at a point, which is called the epicenter of the triangle.

Sagacity theorem: The three heights of a triangle intersect at a point, which is called the sag of a triangle.

Internal Theorem: The bisectors of the three internal angles of a triangle intersect at a point, which is called the heart of the triangle.

Proximity theorem: the bisector of an inner angle of a triangle and the bisector of an outer angle of the other two vertices intersect at a point, which is called the proximal point of a triangle. A triangle has three side centers. The center of gravity, the outer center, the vertical center, the inner center and the lateral center of a triangle are called the five centers of the triangle, and they are all important connection points of the triangle. The above conclusions were discovered as early as Euclid's time, and Euclid collected them as important theorems except the vertical center theorem in his Elements of Geometry. Physical term definition of center of gravity: all parts of an object are acted by gravity. From the effect point of view, we can think that the gravity of each part is concentrated on one point, which is called the center of gravity of the object. The position of the center of gravity of an object is an object with uniform mass distribution (uniform object), and the position of the center of gravity is only related to the shape of the object. For objects with regular shapes, the center of gravity is on the geometric center of gravity. For example, the center of a uniform thin straight rod is at the midpoint of the rod, the center of gravity of a uniform sphere is at the center of the sphere, and the center of gravity of a uniform cylinder is at the midpoint of the shaft. The center of gravity of an irregular object can be determined by hanging method, and the center of gravity is not necessarily on the object. For objects with uneven mass distribution, the position of the center of gravity is not only related to the shape of the object, but also related to the mass distribution inside the object. The center of gravity of the truck changes with the change of loading capacity and loading position, and the center of gravity of the crane changes with the weight and height of the hoisted object.

An object or figure is divided into two parts by the straight line or tangent plane of the center of gravity, so the volumes or areas of the two parts are not necessarily equal. Not all straight lines or tangent planes passing through the center of gravity bisect the area or volume of an object or figure. For example, a straight line passing through the center of gravity of a regular triangle and parallel to one side divides the triangle into two parts with an area ratio of 4: 5. This can be explained by the lever principle of physics: the distance from the center of gravity of two divided figures to the center of gravity of a triangle is equivalent to the two arms of the lever, and the area of the two figures is equivalent to the two forces of the lever. Because the center of gravity is equivalent to the point where two graphic areas are "concentrated" (refer to the definition of the center of gravity). Like the above example, the distance between the center of gravity of the two divided figures and the center of gravity of the triangle is exactly equal to 5: 4. If you are interested, you can prove it by drawing with geometric sketchpad software. ) Mathematical method for determining the position of the center of gravity of an object: If any definite rectangular coordinate system O-xyz is selected in the space where an object (with a total mass of m) is located, the object can be subdivided into I particles, and each particle corresponds to its own coordinate (xi, yi, zi) and mass mi, so it is easy to know that M=m 1+m2+‥+mi.

Then x = (x1m1+x2 m2+‥+ximi)/m.

y=(y 1m 1+y2m2+‥+yimi)/m

The function of z = (z1m1+z2m2+‥+zimi)/m The ordinary people have limbs and trunk. Head first. It stands and throws the ball. And posture. Stand straight. The center of gravity was born. Center of gravity is stable. The so-called opportunity to build momentum. The center of gravity is gone. There is a danger of reversal. That is, there is no chance. It is out of the question. Boxing is the basis of function. It depends on the stability of the center of gravity. And the center of gravity can be divided into fixed and mobile. Fixer. Master, it's time to practice boxing by yourself. Every move. A gesture. We must always pay attention to it. Or turn around. Or both. Center of gravity and virtual reality belong to one. Reality can change impermanence The center of gravity is not. Can move. Because it is the master of everything. Don't make a move. Let the enemy know whether we are false or true. Another example is fighting. Heart is order. Qi is a flag. The waist is a squat. Tai Ji Chuan takes strength as a tactic. Virtual reality is a strategy. Spirit is command. Listening is a rare thing. Focus on the coach. Scholars. We should always feel it silently. This will be of great use to the whole of Stowe. The center of gravity activity is called. This is between me and him. Although in a duel. You must always keep your center of gravity. And attack the center of gravity of others. That is, adhere to the commander-in-chief of the whole army. I won't let the coach lose. The center of gravity of a triangle is the intersection of three sides of the triangle, and the intersection of three lines can be proved by the dovetail theorem, which is very simple. The proof process is a special case of Seva theorem.

Several properties 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.

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, its coordinate is ((x 1+x2+x3)/3, (y1+y2+y3)/3); Spatial rectangular coordinate system-abscissa: (X 1+X2+X3)/3 ordinate: (Y 1+Y2+Y3)/3 ordinate: (z 1+z2+z3)/3.

5. The point where the product of the distance from a triangle to three sides is the largest. The center of gravity of the line segment is the center of the line segment. The center of gravity of a parallelogram is the intersection of its two diagonals. The influence factors of the center of gravity are 1 and the shape of the object.

2. Method of finding the center of gravity by mass distribution A. Hanging method

Only suitable for very thin objects. First find a thin rope, find a point on the object, hang it with a rope, and draw the gravity line of the object after it is still. Similarly, find a point to hang it. The intersection of two gravity lines is the center of gravity of the object. B, support methods

What supports an object is a point, and its position is constantly changing. The more stable the position, the closer to the center of gravity.

Properties of triangle center of gravity

The center of gravity is the intersection of three sides of a triangle, 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. Equal product: the areas of the three triangles composed of the center of gravity and the three vertices of the triangle are equal.

3. The sum of squares of the distances from the center of gravity to the three vertices of the triangle is the smallest.

4. The vector sum from the center of gravity to the three vertices of the triangle is zero.