What are the five centers of a triangle?

The center of gravity, external center, vertical center, internal center and transverse center of a triangle are called the five centers of the triangle ... The five-center method of the triangle refers to the general term of the center of gravity method, external center method, vertical center method, internal center method and transverse center method of the triangle.

(1) Law of the center of gravity of a triangle: The center lines of three sides of a triangle intersect at a point, which is called the center of gravity of the triangle. The intersection of three lines can be proved by dovetail theorem, which is very simple.

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 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.

4. In the plane rectangular coordinate system, the coordinate of the center of gravity is the arithmetic average of the vertex coordinates, that is, the coordinate of the center of gravity is [(x 1+x2+x3)/3], [y 1+y2+y3/3]].

(2) The Law of the Exterior Center of the Triangle: The perpendicular lines of the three sides of the triangle intersect at one point. This point is called the outer center of the triangle. In other words, a triangle is the center of a tangent circle. Note that the distances from the outer center to the three vertices of the triangle are equal, and the outer center theorem combined with the definition of the middle vertical line is actually well proved.

Before calculating the coordinates of the center of gravity, the following temporary variables should be calculated: d 1, d2 and d3 are the point multiplication of three vertices of a triangle connected with the other two vertex vectors respectively. 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).

(3) The law of the vertical center of a triangle: the three heights of a triangle intersect at a point, which is called the vertical center of a triangle.

The essence of the heart:

1. Six four-point circles can be obtained from three vertices, three vertical feet and seven vertical centers of a triangle.

2. Look outside the heart, look inside.

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.

This point is divided into two parts of each high line of the product.

Legal proof

It is known that in Δ δABC, AD and BE are two heights. The intersection of AD and BE connects CO at the point and extends AB at the point F. The proof of the intersection is 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!

(4) The internal law of triangle: the three bisectors of the internal angle of triangle intersect at a point, which is called the inner point of triangle, that is, the center of the inscribed circle of triangle. Note that the distance from the inner heart to the three sides is equal (the radius of the inscribed circle), and the inner heart theorem is actually easy to prove.

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.

(5) Proximity law of triangle: the bisector of the inner angle and the bisector of the outer angle of triangle intersect at one point 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.

Triangle Five-Heart Song (Heavy Hanging Outside and Inside)

A triangle has five hearts; It is very important to attach importance to the outside, the inside and the side, so it is very important to master it carefully.

Chongxin

The three midlines must intersect, and the location of the intersection 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. Let three sides be perpendicular to each other and three lines intersect at one point.

This point is defined as the epicentre and can be used as the circumscribed circle.

worry (about/over)

If a triangle is three high points, the three high points must intersect at the vertical center. The high line is divided into triangles with three pairs of right angles.

There are twelve right triangles, forming six pairs of similar shapes, which can be found in the four-point * * * diagram. Careful analysis can clearly find them.

heart

A triangle corresponds to three vertices, each corner has a bisector, and the three lines intersect at a certain point, which is called "inner heart" and has roots;

The points to the three sides are equidistant and can be inscribed into a triangle.