Center of gravity-the intersection of three midlines of a triangle. The center of gravity divides the length of the center line into 2:1;
Vertical center-the intersection of three perpendicular lines of a triangle, and the cross product of the perpendicular line and the corresponding side is 0;
Inner heart-the intersection point of the bisector of the three internal angles of a triangle (the center of the inscribed circle of the triangle), and the distance from any point on the bisector to both sides of the angle is equal;
Extended data
First, the height of the triangle
(1) Definition: Draw a vertical line from a vertex of a triangle to the line where the opposite side is located, and the line segment between the vertex and the vertical foot is called the height of this side.
(2) Drawing language: ad ⊥ BC with point A as point D (D is vertical foot and AD is vertical segment).
(3) Inference language: ∵AD is the height of △ABC ∴AD⊥BC (or ∠ ADB = ∠ ADC = 90).
(4) Vertical center: The intersection of straight lines of three heights of a triangle is called the vertical center.
(5) Height and vertical center of different triangles
Acute triangle: all three heights are inside the triangle, and the vertical center is also inside;
Xiehe line: two heights are outside the triangle, one is inside the triangle, and the center is outside;
Right triangle: two heights coincide with the right-angled side, the other is inside the triangle, and the vertical center is the right-angled vertex.
(6) Area of triangle: the area formula of triangle: S= 1/2 base × height.
Second, the center line of the triangle
(1) Definition: In a triangle, a vertex is connected with the midpoint of its opposite side, and the resulting line segment is called the midline of the side.
(2) Drawing language: take the midpoint D of BC side and connect with AD.
(3) Reasoning language
∵AD is the center line of △ABC.
∴BD=DC= 1/2BC
(or BC=2BD=2DC)
(4) Center of gravity: The intersection of the three midlines of a triangle is called the center of gravity.
(5) The midline and center of gravity of different triangles: the midline and center of gravity of all triangles are inside the triangle.
(6) the nature of the 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.
Take mid-line advertising as an example, and everything else holds.
AO:OD=2: 1
AO = 2OD
OD= 1/2AO
OD= 1/3AD
AO=2/3AD
(2) The areas of the three triangles formed by the center of gravity and the three vertices of the triangle are equal.
Namely: area of △AOB = area of △ AOC = area of △ BOC.
(The proof method uses the nature of gravity ①, and interested students can try to prove it. )
Third, the bisector of the triangle.
(1) Definition: The bisector of an angle of a triangle intersects its opposite side, and the line segment between the vertex and the intersection of this angle is called the bisector of the triangle.
(2) Drawing language: Let the bisector AD whose point A is ∠BAC intersect BC at point D..
(3) Reasoning language
∫AD is the angular bisector of △ABC.
∴∠ 1=∠2= 1/2∠BAC
(4) Inner heart: The intersection of the bisectors of the three angles of a triangle is called the inner heart.
(5) The bisectors and centers of different triangles: The bisectors and centers of all triangles are inside the triangle.
(6) The nature of the heart: the distance from the heart to each side of the triangle is equal.
Baidu Encyclopedia-Four Hearts of Triangle