Have time to teach you how to remember.
Triangle center problem:
Gravity center theorem: Three midlines of a triangle intersect at a point, which is the vertex.
The distance is twice the distance from the opposite midpoint. This point is called the center of gravity of the triangle.
Eccentricity theorem: the perpendicular lines of three sides of a triangle intersect at one point. This point is called the outer center of the triangle.
Vertical center theorem: three heights of a triangle intersect at one point. This point is called the center of the triangle.
Interior Theorem: The bisectors of three interior angles of a triangle intersect at one point. This point is called the heart of a triangle.
Proximity theorem: the bisector of the inner angle of a triangle intersects the bisector of the outer angle of the other two vertices at one point. This point is called the center of the triangle. A triangle has three side centers.
Several important theorems about the center of gravity of triangle
1. The center of gravity is the intersection of three sides of a triangle, and 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.
The nature of the external world:
1, if O is the epicenter of △ABC, then ∠BOC=2∠A(∠A is an acute angle or a right angle) or ∠ BOC = 360-2 ∠ A (∠ A is an 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 coincides with the midpoint of the hypotenuse.
4. To calculate the coordinates of the center of gravity, we must first calculate the following temporary variables: d 1, d2, d3 are the point multiplication of three vertices of a triangle connected with the other two vertex vectors. c 1=d2d3，c2=d 1d3，C3 = d 1 D2； C=c 1+c2+c3. Coordinate of gravity center: ((c2+c3)/2c, (c 1+c3)/2c, (c 1+c2)/2c).
The essence of the heart:
1, three vertices and three vertical feet of a triangle, and these seven points can get six four-point circles.
2. The triangle three-point * * line of the outer center O, the center of gravity G and the vertical center H, OG∶GH= 1∶2. This line is called the Euler line of triangle. )
3. The distance from the vertical center to the vertex of the triangle is twice as long as the distance from the outer center of the triangle to the opposite side of the vertex.
The product of two parts of each high line is equal.
Legal proof
It is known that in Δ δABC, AD and BE are two heights, which intersect at point O, connect CO and extend the intersection of AB at point F. Verification: CF⊥AB.
Prove:
Connect de≈ADB =∠aeb = 90 degrees ∴A, B, D and E * * * cycles ∴∠ADE=∠ABE.
∵∠eao=∠dac∠aeo=∠adc∴δaeo∽δadc
∴ae/ao=ad/ac∴δead∽δoac∴∠acf=∠ade=∠abe
∠∠Abe+∠BAC = 90 degrees ∴∠ACF+∠BAC=90 degrees ∴∴ CF ⊥ AB.
Therefore, the law of hanging heart holds!
The bisectors of the three internal angles of a triangle intersect at one point. This point is called the heart of the triangle, which is the center of the inscribed circle of the triangle. Note that the distance from the center to the three sides is equal (the radius of the inscribed circle), and the internal law is actually easy to prove.
Nature:
If the three sides are l 1, l2, l3, and the perimeter is p, then the coordinates of the center of gravity of the heart are (l 1/p, l2/p, l3/p).
The distance from the center to the side of a right triangle is equal to the sum of two right angles minus half the difference of the hypotenuse.
The projection of the center of a triangle consisting of a point and two focal points on any branch of a hyperbola on the real axis is the vertex of the corresponding branch.
There are 1 important attributes:
The bisector of the inner corner of a triangle intersects the bisector of the outer corner at the other two vertices. The center of the tangent circle of a triangle (the circle tangent to the extension line of one side and the other two sides of the triangle) is called the tangent center.
Nature:
Each triangle has three side centers.
It is equidistant from three sides.
As shown in the figure, point m is the centroid of △ABC. The intersection of the bisector of the outer angle of any two angles of a triangle and the bisector of the inner angle of the third angle. A triangle has three side centers, and it must be outside the triangle.
Attachment: the center of a triangle: only a regular triangle has a center. At this time, the center of gravity, inner heart, outer heart, hanging heart and four hearts are integrated.
etc