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.

5. The center of gravity is the point where the product of the distances from the triangle to the three sides is the largest.

6. The center of gravity of triangle ABC is G, and point P is any point inside it, so 3PG? =(AP？ +BP？ +CP？ )- 1/3(AB？ +BC？ +CA？ )。

7. In the triangle ABC, if the straight line passing through the center of gravity G intersects with the straight lines where AB and AC are located at P and Q respectively, AB/AP+AC/AQ=3.

8. Tangents are made from the three vertices of the triangle ABC to the circle with the opposite side as the diameter, and the six tangent points obtained are Pi, then Pi is centered on the center of gravity G, and r =118 (ab? +BC？ +CA？ ) on the circumference of the radius.

9.g is the center of gravity of the triangle ABC, and P is any point on the plane of the triangle ABC, so PA? +PB？ +PC？ =GA？ +GB？ +GC？ +3PG？ .

Extended knowledge: barycenter coordinates

In mathematics, the coordinates of the center of gravity are defined by the vertices of a simplex (such as a triangle or tetrahedron). The barycenter coordinate is a kind of homogeneous coordinate.

Calculation method of barycenter coordinates:

For objects with regular shapes, such as spheres and cubes, the center of gravity is located at the geometric center. For objects with irregular shapes, it can be calculated by centroid method, gravity center method or integral method.

Centroid method: the centroid is the average position of each part of the object, which is obtained by calculating the sum of the products of the mass of each part of the object and the corresponding position, and then dividing by the total mass of the object. The centroid method is suitable for objects with uniform mass distribution.

Gravity center method: the gravity center is the position where all parts of the object pass through by the combined force of gravity. For a uniform thin rod, the center of gravity is located at the midpoint of the rod; For uniform thick plates, the center of gravity is located in the center of the plate. For irregular objects, the center of gravity can be determined by hanging method, that is, hanging the object on a fulcrum to make the object reach balance, and the position of the fulcrum is the center of gravity.

Integration method: For objects with complex shapes, the position of the center of gravity can be obtained by integrating the mass density function on the surface or in the body of the object.