Talk about center of gravity and center

The point where the center of gravity, vertical center, outer center and inner center of a regular triangle coincide is called the center.

All parts of an object are affected 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.

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 distances from a triangle to three sides is the largest.

The intersection of three heights of a triangle is called the vertical center of the triangle.

The vertical center of an acute triangle is inside the triangle.

The vertical center of a right triangle is at the vertex of the triangle.

The vertical center of an obtuse triangle is outside the triangle.

The vertical center is the intersection of high lines.

The vertical center is the intersection of three vertical lines from each vertex of a triangle to its opposite side.

Three vertices, three vertical feet and seven vertical centers of a triangle can get six four-point circles.

The heart is the intersection of bisectors of three internal angles of a triangle, that is, the center of the inscribed circle.

The distance from the center to the side of a right triangle is equal to the sum of two right angles minus half the difference of the hypotenuse.

The epicenter is the intersection of the perpendicular bisector of the three sides of the triangle, which is the center of the circumscribed circle.

The center of the tangent circle of a triangle (the circle tangent to the extension line of one side and the other two sides of the triangle) is called the tangent center. Paracenter is the intersection of the interior bisector of a triangle and its two non-adjacent exterior bisectors, and its distances to the three sides are equal. . The intersection of the bisector of the outer angle of any two angles of a triangle and the bisector of the inner angle of the third angle. A triangle has three side centers, and it must be outside the triangle.