A summary of the knowledge about the triangle inner inner outer center of gravity vertical heart and

Not necessarily. The center of gravity, the heart, is inside the triangle. The center of gravity is the intersection of the three middle lines of the triangle, and the heart is the intersection of the three angular bisectors of the triangle, and both the midline and the angular bisector are inside the triangle.

The outer center is the perpendicular bisector of the three sides of the triangle.

The intersection of the outer center of an acute triangle is inside the triangle, a right triangle.

The outer center of the obtuse triangle is at the midpoint of the hypotenuse, while the outer center of the obtuse triangle is on the outside of the triangle.

The paracenter is the center of the circumscient circle of a triangle, and a triangle has three paracenters, all of which are on the outside of the triangle.

Orthocenter. It is the intersection of the three heights of the triangle. The vertical center of an acute triangle is inside the triangle, the vertical center of a right triangle is at the vertex of the right angle, and the vertical center of an obtuse triangle is on the outside of the triangle.

I hope I can help you with your doubts.

The center of gravity and the heart must be inside the triangle;

The outer center and perpendicular center are not necessarily inside the triangle, but may also be on or outside the triangle;

The center must be outside the triangle.

Not necessarily, only the center of gravity and the heart must be inside the triangle.

Not necessarily, a regular triangle is like this, an acute angle is also true, and an obtuse angle is not.

1. The intersection point of the three perpendicular lines of the triangle is called the outer center, that is, the center of the outer circle.

2. The intersection of the bisector of the three corners of the triangle is called the inner part of the triangle, that is, the center of the inscribed circle.

3. The intersection of the three heights of the triangle is called the vertical center.

4. The intersection of the three middle lines of the triangle is called the center of gravity.

5. Only when the triangle is a regular triangle, the center of gravity, vertical center, inner center, and outer center are united into one heart, which is called the center of the regular triangle.

Triangle vertical center definitionThe vertical center is the intersection of three perpendicular lines made from each vertex of the triangle to its opposite edge.

An acute triangle is centered inside the triangle.

The right triangle is centered vertically at the right-angled vertex of the triangle.

The obtuse triangle is perpendicular to the outside of the triangle.

The triangle has three vertices, three vertical feet, and the seven vertical centers can give 6 groups of four-point contours.

The center of gravity, the outer center, the vertical center, the inner heart and the side center of the triangle are called the five hearts of the triangle.

The five-heart theorem of triangles refers to the general term of the triangle's center of gravity theorem, outer center theorem, perpendicular theorem, inner theorem, and side center theorem.

Center of gravity: The midline of the three sides of the triangle intersects at one point. This point is called the center of gravity of the triangle.

Outer center: The center of the triangle is called the outer center of the triangle.

Perpendicular: The three heights of the triangle (the straight line) intersect at one point, which is called the vertical center of the triangle.

Heart: The center of the triangle is cut into the circle, and the heart of the triangle is called to imitate the brother Hui.

Circumcentric: The center of the circumcentric circle of the triangle (the circle tangent to one side of the triangle and the extension lines on the other two sides) is called the circumcentric Zheng Chenxian of the triangle.

The triangle has only 5 hearts: inner and outer hearts.

The center of gravity, the core of the forest, the heart, the heart.

They are: the intersection of the bisector of the inner angle (the center of the inscribed circle), and the perpendicular line.

Intersection (circumscribed circle.

center of the circle), the intersection of the middle line (hanging equilibrium point), the intersection of the inner bisector and its two outer bisectors that are not adjacent to it (the center of the circumscribed circle), the intersection of three high lines.

As for the "center", it is not the "heart" of the general triangle, because not every triangle has a center, only the special three-do clear angle before the pure family, such as the "equilateral triangle".

The center of the triangle: Only when the triangle is a regular triangle, the center of gravity, the vertical center, the inner center, and the outer center are united into one center, which is called the center of the regular triangle.

The center of gravity of the triangle: the intersection of the three midlines, which is twice the distance from the vertex to the midpoint of the opposite side. The center of gravity to the median ratio is 1:2.

The heart of the triangle: the intersection of the three angular bisectors, which is the abbreviation for the center of the inscribed circle of the triangle. The distance to the three sides is equal.

The outer center of the triangle: the intersection of the three perpendicular lines, which is the abbreviation of the center of the circumscribed circle of the triangle. The distance to the three vertices is equal.

Please click Enter a description.

Please click Enter a description.

Extended Materials. Nature of the center of gravity:

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.

The area of the three triangles formed by the center of gravity and the three vertices of the triangle is equal. That is, the distance from the center of gravity to the three sides is inversely proportional to the length of the three sides.

The sum of squares of the distance from the center of gravity to the 3 vertices of the triangle is minimum.

In a planar Cartesian coordinate system, the coordinates of the center of gravity are the arithmetic mean of the vertex coordinates, i.e., the coordinates of its center of gravity are ((x1+x2+x3) 3,(y1+y2+y3) 3.

Please click Enter a description.

The center of the triangle is called the outer center of the triangle The properties of the outer center:

The perpendicular bisector of the three sides of a triangle intersects at a point where the triangle is the outer center of the triangle.

If O is the outer center of ABC, then BOC = 2 A (A is an acute or right angle) or BOC = 360°-2 A (A is an obtuse angle).

When the triangle is an acute triangle, the outer center is inside the triangle; When the triangle is obtuse, the outer center is outside the triangle; When the triangle is a right-angled triangle, the outer center is on the hypotenuse, which coincides with the midpoint of the hypotenuse.

To calculate the coordinates of the center of gravity of the outer center, the following temporary variables should be calculated: d1, d2, d3 are the dot times of the three vertices of the triangle to the other two vertex vectors, respectively. c1=d2d3，c2=d1d3，c3=d1d2；c=c1+c2+c3。

Center of gravity coordinates: ( C2+C3) 2C, (C1+C3) 2C, (C1+C2) 2C).

The distance from the outer center to the three lead field points is equal to the center of the triangle inscribed circle, and the number of volumes is called the inner part of the triangle The nature of the heart:

The three inner bisectors of the triangle intersect at one point. This point is the heart of the triangle.

The distance from the inside of a right triangle to the side is equal to one-half of the difference between the two right sides and minus the hypotenuse.

p is any point on the plane where δabc is located, and the point i is the sufficient and necessary condition of δabc is that the vector pi = (a vector pb+b vector pb+c vector pc) (a+b+c).

The outer center is the center of the circumscribed circle, at this time the three vertices of the triangle are on the circle, and the distance from the center of the circle to the three vertices is equal, that is, the distance from the outer center to the three vertices of the triangle is equal, so the outer center is the intersection of the perpendicular lines of the three sides of the triangle.

At this time, the three sides of the triangle are tangent to the circle, and the distance from the center of the circle to the three sides is equal, that is, the distance from the heart to the three vertices of the triangle is equal, so the heart is the intersection of the angular bisector of the three corners of the triangle.

The center of gravity is the intersection point of the three middle lines, which are connected to the midpoint of the opposite side through three vertices, the intersection point of the center line is the center of gravity, and the center of gravity divides the three center lines into 1:2, that is, the distance between the center of gravity and the midpoint and the distance between the center of gravity and the vertex ratio is 1:2

The vertical center is the intersection of the three high points, and the vertical line is made by the opposite side of the three vertices, and the intersection of the vertical line is the vertical hail Zheng Xin.

The content is estimated to be incomplete, so let's wait for a master.

A triangle has only five kinds of hearts.

Center of gravity: the intersection point of the three midlines, the three midlines of the triangle intersect at one point, and the distance from this point to the vertex is twice the distance from it to the midpoint of the opposite side; The center of gravity to the median ratio is 1:2.

Perpendicular: the intersection of the three heights of the triangle;

Heart: The intersection of the bisector of the three inner angles, which is the abbreviation of the center of the inscribed circle of the triangle;

The distance to the three sides is equal.

Outer center: the intersection point of the three perpendicular lines, which is the abbreviation of the center of the circumscribed circle of the triangle; Equal distance to three vertices: the intersection of one of the inner bisectors with the other two outer bisectors. (There are three of them.) ) is an abbreviation for the center of a circumscribed circle of a triangle.

If and only if the triangle is a regular triangle, the four centers are united into a center, which is called the center of the regular triangle.

The center of gravity is the intersection of the three sides of the triangle.

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, and the area of the three triangles formed by the three vertices of the triangle is equal to the area.

The sum of squares of the distance from the center of gravity to the 3 vertices of the triangle is minimum.

In a planar Cartesian coordinate system, the coordinates of the center of gravity are the coordinates of the vertices, and the intersection of the three heights of the arithmetic mean triangle is called the vertical center of the triangle.

An acute triangle is centered inside the triangle.

The right triangle is centered vertically at the right-angled vertex of the triangle.

The obtuse triangle is perpendicular to the outside of the triangle.

The inner is the intersection of the three bisectors of the inner angles of the triangle, that is, the center of the inscribed circle.

The distance from the inner to the three sides is equal (the radius of the inscribed circle).

If the three sides are L1, L2, L3, and the circumference is P, then the center of gravity coordinates of the heart are (L1 P, L2 P, L3 P).

The distance from the inside of a right triangle to the side is equal to one-half of the difference between the two right sides and minus the hypotenuse.

The projection of the inner part of the triangle formed by the upper point and the two foci on the hyperbola on the real axis is the vertex of the corresponding branch.

The outer center is the intersection of the perpendicular bisector of the three sides of the triangle, i.e., the center of the circumscribed circle.

The three vertices to the outer center to the triangle are at equal distances.

Heart: the intersection of the bisector of the three inner angles;

Outer center: the intersection of the three perpendicular bisectors;

Center of gravity: the intersection of the midline on three sides;

Perpendicular: The intersection of the three high sides.

Center: [This is only found in a regular triangle, and at this point, the above hearts coincide].

Center of gravity: Triangle.

The intersection of the three middle lines.

Perpendicular: The intersection of the three heights of the triangle at the intersection of the straight lines.

Outer center: The center of the triangle circumscribed by the circle, that is, the three-sided vertical bisector.

of the intersection. Heart: The center of the inscribed circle of the triangle, i.e., the intersection of the bisector of the three angles.

Inner: the intersection of the perpendicular lines on each side, the outer center: the intersection of the bisector of each angle, the center of gravity in the coordinate system, the vertical center:

The intersection of the perpendicular lines from the corners to the opposite side!

The inner is the center of the inscribed circle, located at the intersection of the bisector of the angle.

The outer center is the center of the inscribed circle, which is located at the intersection of the perpendicular lines on each side.

The center of gravity is the point where gravity acts on a uniform geometry and is located at the intersection of the midlines of each side.

The center of the vertical is the intersection of the high.