What are the centers of a triangle

What are the centers of a triangle

The so-called "four centers" of a triangle refer to four special points formed by the intersection of four important line segments of a triangle. They are the inner center, outer center, perpendicular center and center of gravity of a triangle
1. Center
The heights of the three sides of a triangle meet at a point called the perpendicular of the triangle
2. Center of gravity
The center lines of the three sides of a triangle meet at a point, which is called the center of gravity of the triangle
3. The intersection of the middle and vertical lines of the three sides of the triangle is at a point, which is called the center of the circumscribed circle of the triangle
4. The bisectors of the three inner angles of a triangle meet at a point, which is called the center of the inscribed circle of the triangle,
The intersection of the center lines on the three sides of the center of gravity
The intersection of three heights perpendicular to the center
The intersection of the bisectors of the three corners of the inner inscribed circle
The intersection of the vertical bisectors of the three sides of the center of a circumscribed circle
There is also a center called paracenter: the intersection of the bisector of the outer corner (there are three), or the center of the circumscribed circle (or the center of the circumscribed circle) has only the center of an equilateral triangle. At this time, the center of gravity, the inner center, the outer center, the perpendicular center, and the four centers are in one

How to judge the shape of Zheng! (except definition)

Corresponding to the definition of rectangle, Zheng shape is defined as two groups of quadrilateral whose adjacent sides are equal
The second definition of Zheng shape: a quadrilateral with one diagonal dividing vertically and the other diagonal is Zheng shape
Obviously, diamond is a special Zheng shape
The character of Zheng shape:
1. Axisymmetric, the axis of symmetry is a diagonal of Zheng shape
2. There is a group of diagonally equal. For the convenience of discussion, we might as well call this group of diagonally equal
3. Area formula of Zheng shape
S = Mn / 2, where m and N are the lengths of two diagonals
S = absin a, where a and B are opposite sides of Zheng shape, and a is equiangular of Zheng shape
S = (a ^ 2sinb + B ^ 2sinc) / 2, where B and C are a set of unequal angles of Zheng shape
4. The circumference formula of Zheng shape: C = 2 (a + b)
5. Zheng shape has inscribed circle. The center of the inscribed circle is the intersection of Zheng symmetry axis and equiangular bisector
6. The necessary and sufficient conditions for Zheng to have circumcircle are as follows
2Ab = Mn or a = 90 degrees or B + C = 180 degrees
7. The line between the inscribed circle of Zheng shape and the four tangent points of the four sides is isosceles trapezoid. The line between the inscribed circle of Zheng shape and the four intersection points of the two diagonal lines is still Zheng shape