(1) Definition: The center of a triangle is the intersection of three bisectors of a triangle or the center of an inscribed circle.

(2) The inherent nature of triangle

① The bisectors of the three angles of a triangle intersect at one point, which is the heart of the triangle.

(2) The distance from the center of the triangle to the three sides is equal, which is equal to the radius r of the inscribed circle.

③s=(r is the radius of inscribed circle)

④ in Rt△ABC, ∠ c = 90, r = (a+b-c)/2.

⑤∠BOC = 90+∠A/2∠BOA = 90+∠C/2∠AOC = 90+∠B/2

2. Outside the heart

(1) Definition: The outer center of a triangle is the intersection of three perpendicular bisector of the triangle (or the center of the circumscribed circle of the triangle).

(2) the nature of the triangle's outer center

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. The distances from the outer center to the three vertices are equal.

3. Center of gravity

(1) The median lines of three sides of a triangle intersect at one point. This point is called the center of gravity of the triangle. (2) the nature of the triangle center of gravity

① 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 Any connecting line between the center of gravity and the three vertices of the triangle divides the triangle area equally.

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

4. Hold on to your heart

(1) Definition: The vertical center of a triangle is the intersection of the heights of three sides of the triangle (usually expressed by H).

(2) The nature of the vertical center of triangle

① The vertical center of the acute triangle is within the triangle; The vertical center of a right triangle is at the right vertex; The vertical center of an obtuse triangle is outside the triangle.

② The vertical center of a triangle is the center of its vertical triangle; In other words, the center of a triangle is the center of the triangle next to it.

③ The symmetrical points of the vertical centers of the left and right sides are all on the circumscribed circle of △ABC.

Extended data:

The distance from any vertex of a triangle to the vertical center is equal to twice the distance from the outer center to the opposite side.

The sum of the distances from the vertical center to the three vertices of an acute triangle is equal to twice the sum of the radii of its inscribed circle and circumscribed circle.

Gravity center memory formula:

The three midlines must intersect, and the location of the intersection is really strange. The intersection is named "center of gravity", and the nature of the center of gravity should be clear.

External memory formula:

A triangle has six elements, three inner corners and three sides. Let three sides be perpendicular to each other and three lines intersect at one point.

Careless memory formula:

The angle is three highs, and the three highs must intersect at the vertical center. The high line is divided into triangles with three pairs of right angles.

There are twelve right triangles, forming six pairs of similar shapes, which can be found in the four-point * * * diagram. Careful analysis can clearly find them.

Memory formula:

A triangle corresponds to three vertices, each corner has a bisector, and the three lines intersect at a certain point, which is called "inner heart" and has roots;

The points to the three sides are equidistant and can be inscribed into a triangle. This center of the circle is called "inner heart", so it is natural to define it.

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