What needs to be added is that triangles also have side centers. The center of gravity, the outer center, the hanging center, the inner center and the lateral center of a triangle are usually called the five centers of the triangle.
First, the triangle center of gravity theorem
The median lines of three sides of a triangle intersect at one point. This point is called the center of gravity of the triangle.
The intersection of three median lines at one point can be proved by the dovetail theorem, which is very simple.
(The center of gravity was originally a physical concept. For a triangular thin plate with equal thickness and uniform mass, its center of gravity is exactly the intersection of the three midlines of this triangle, hence the name. )
Nature of the center of gravity:
1, 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 center of gravity is equal to the area of the three triangles formed by any two vertices of the triangle. That is, the distance from the center of gravity to the three sides is inversely proportional to the growth of the three sides.
3. The sum of 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 average of the vertex coordinates.
That is, its barycenter coordinates are ((X 1+X2+X3)/3, (Y 1+Y2+Y3)/3. Third, the triangle vertical center theorem
The three heights of a triangle intersect at a point, which is called the vertical center of the triangle.
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. Triangular three-point * * line of outer center O, center of gravity G and 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.
theorem proving
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 vertical center theorem holds! Fourthly, the interior theorem of triangle.
The center of the inscribed circle of a triangle is called the heart of the triangle.
Intrinsic essence:
1, the three bisectors of the triangle intersect at one point. This point is the center of the triangle.
2. The distance from the center to the right-angled triangle edge is equal to the sum of the two right-angled edges minus half the difference of the hypotenuse.
3.p is any point on the plane of Δ ABC, and point 0 is the inner heart of Δ ABC if and only if:
Vector P0=(a× vector PA+b× vector PB+c× vector PC)/(a+b+c).
4.o is the heart of the triangle, and A, B and C are the three vertices of the triangle. If BC edge of AO intersection extends to n, there will be
AO:ON=AB:BN=AC:CN=(AB+AC):BC
5. The necessary and sufficient conditions for point O to be any point on the plane ABC and point I to be △ABC are as follows:
A (vector OA)+b (vector OB)+c (vector OC)= vector 0.
6. (euler theorem) ⊿ABC, where R and R are the radii of the circumscribed circle and the inscribed circle, and O and I are the outer center and the inner center, respectively, then Oi 2 = R 2-2rr.
7. (The bisector of the inner corner is divided into three sides)
In △ABC, 0 is the center, and the bisectors of internal angles of ∠A, ∠B and ∠C intersect BC, AC and AB at Q, P and R respectively.
Then bq/QC = c/b, CP/pa = a/c, br/ra = a/B. Second, the triangle eccentricity theorem.
The center of the circumscribed circle of a triangle is called the outer center of the triangle.
The nature of the external world:
1. The perpendicular bisector of three sides of a triangle intersect at a point, which is the outer center of the triangle.
2. If O is the outer center of △ABC, ∠BOC=2∠A(∠A is acute angle or right angle) or ∠ BOC = 360-2 ∠ A (∠ A is 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 is on the hypotenuse and coincides with the midpoint of the hypotenuse.
4. To calculate the coordinates of the epicenter, we must first calculate the following temporary variables: d 1, d2 and d3 are the point multiplication of the vectors whose three vertices are connected with the other two vertices.
c 1=d2d3，c2=d 1d3，C3 = d 1 D2； c=c 1+c2+c3 .
Eccentric coordinates: ((c2+c3)/2c, (c 1+c3)/2c, (c 1+c2)/2c).
5. The distances from the outer center to the three vertices are equal. 5. Triangle Proximity Theorem
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 edge center of the triangle.
Nature of lateral center:
1, the bisector of an inner corner of a triangle and the bisector of an outer corner of the other two vertices intersect at a point, which is the edge center of the triangle.
2. Every triangle has three side centers.
3. The distance from the side center to the three sides is equal.
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. A poem about the five hearts of a triangle: a song of the five hearts of a triangle (focusing on the outside and the inside)
A triangle has five hearts, one hanging outward and the other hanging inward.
The nature of the five minds is very important, so it is important to master the confusion carefully and concentrate on the mind.
The three midlines must intersect, and the intersection position is really strange.
The intersection is named "center of gravity", and the nature of the center of gravity should be clear.
In the center of gravity segmentation, you can hear the ratio of line segment to several segments;
The ratio of length to length is two to one, so we should use it flexibly and master it well.
External heart
A triangle has six elements, three inner corners and three sides.
Make the middle of the three sides vertical and the three lines intersect at one point.
This point is defined as the epicenter and can be used as the circumscribed circle.
Don't forget the inner and outer heart, the key is to cut in and cut out.
worry (about/over)
If a triangle is three high points, then the three high points must cross.
The high line is divided into triangles with three pairs of right angles.
There are twelve right triangles, forming six pairs of similar shapes.
There are four points in the picture, which can be found clearly after careful analysis.
heart
A triangle corresponds to three vertices, and each corner has a bisector.
The intersection of three lines has a * * * point, which is called "inner heart" and has roots;
The points to the three sides are equidistant and can be used as the inscribed circle of a triangle.
This round heart is called "inner heart", so it is natural to define it.
Don't forget the nature of five minds, it's really good to start doing problems.