Who is going to talk about the definition, properties and formulas of the inner center, outer center, center of gravity and perpendicularity of a triangle? be deeply grateful Come on

Who is going to talk about the definition, properties and formulas of the inner center, outer center, center of gravity and perpendicularity of a triangle? be deeply grateful Come on

The center of gravity of a triangle is the point of intersection of its three central lines
The properties of the center of gravity of triangle
1. The ratio of the distance from the center of gravity to the vertex and the distance from the center of gravity to the midpoint of the opposite side is 2:1
2. The area of the three triangles composed of the center of gravity and the three vertices of the triangle is equal
3. The sum of the 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 mean of the vertex coordinates, that is, its coordinate is ((x1 + x2 + x3) / 3, (Y1 + Y2 + Y3) / 3); the space rectangular coordinate system abscissa: (x1 + x2 + x3) / 3, ordinate: (Y1 + Y2 + Y3) / 3, vertical coordinate: (z1 + Z2 + Z3) / 3
5. Any line between the center of gravity and the three vertices of the triangle bisects the area of the triangle
6. The center of gravity is the point where the product of distances from the inside to the three sides of a triangle is the largest
The outer center of a triangle is the intersection of three vertical bisectors of the triangle
Properties of the outer center of a triangle
1. The intersection point of the vertical bisectors of the three sides of a triangle 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 innumerable inscribed triangles of a circle, and the outer centers of these triangles coincide
3. The outer center of an acute triangle is inside the triangle; the outer center of an obtuse triangle is outside the triangle; the outer 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
The center of a triangle is the intersection of its three bisectors
The inner nature of triangle
1. If the bisectors of the three angles of a triangle intersect at a point, that point is the center of the triangle
2. The distance from the center of triangle to three sides is equal, which is equal to the radius of inscribed circle R
3. R = 2S / (a + B + C) 4. In RT △ ABC, 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)
The perpendicular center of a triangle is the intersection of the heights of its three sides (usually represented by H)
The properties of the perpendicular center of triangle
1. The vertical center of an acute triangle is in the triangle; the vertical center of a right triangle is on the right vertex; the vertical center of an obtuse triangle is outside the triangle
2. The perpendicularity of a triangle is the inner part of the triangle, or the inner part of the triangle is the perpendicularity of the triangle
3. The symmetry points of perpendicular o about three sides are all on the circumscribed circle of △ ABC
4. In △ ABC, there are six groups of four points in common circle, three groups of right triangles (four in each group), and Ao · od = Bo · OE = Co · of
5. Any of the four points h, a, B and C is the perpendicular center of the triangle with the other three points as its vertices
6. The circumscribed circles of △ ABC, △ ABO, △ BCO, △ ACO are equal circles
7. In a non right triangle, if the lines AB and AC passing through o are located at P and Q respectively, then AB / AP · tanb + AC / AQ · Tanc = Tana + tanb + Tanc
8. The distance from any vertex of a triangle to the perpendicular is equal to twice the distance from the outer center to the opposite side
9. Let O and H be the outer center and the perpendicular center of △ ABC respectively, then ∠ Bao = ∠ HAC, ∠ ABH = ∠ OBC, ∠ BCO = ∠ HCA. 10. The sum of the distances from the perpendicular 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
11. The center of perpendicularity of an acute triangle is the center of a perpendicular triangle; among the inscribed triangles of an acute triangle (the vertex is on the edge of the original triangle), the perimeter of the perpendicular triangle is the shortest