The four centers of a triangle refer to its center of gravity, outer center, inner center and vertical center. If and only if the triangle is a regular triangle, the center of gravity, center of gravity, inner heart and outer heart are connected into a whole, which is called the center of a regular triangle.

Eccentric nature

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

There is only one circumscribed circle of a triangle, that is, for a given triangle, its outer center is unique, but there are countless inscribed triangles of a circle, and the outer centers of these triangles coincide.

3. The outer center of the acute triangle is in the triangle; The outer center of an obtuse triangle is outside the triangle; The center of a right triangle coincides with the midpoint of the hypotenuse.

4.OA=OB=OC=R

5.∠BOC=2∠BAC，∠AOB=2∠ACB，∠COA=2∠CBA

6.S△ABC=abc/4R

Intrinsic nature

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

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

3.r=2S/(a+b+c)

4. In Rt△ABC, ∠ c = 90, r = (a+b-c)/2.

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

6.S△=[(a+b+c)r]/2 (r is the radius of the inscribed circle)