Can anyone tell me about the relationship between the four hearts of the triangle and the three ve

1 is directly related to your topic, only the outer center (the intersection of the three perpendicular bisectors) has a fixed relationship with the three vertices, i.e. the distance to the three vertices is equal. Center of gravity: The line connecting the three vertices to the center extends to the opposite side, the midline, which must divide the triangle into two parts of equal (not necessarily congruent).

2 digressions. The inner (intersection of the three inner bisectors) and the center of gravity (the intersection of the three midlines) must be within the triangle, whether obtuse or not. The outer center and the vertical center (the intersection of the three heights) are not necessarily in the form.

Outer center: The acute triangle must be inside the shape, the right triangle is at the midpoint of the hypotenuse, and the obtuse triangle is outside the shape. Vertical Heart:

Acute triangles must be inside the shape, right triangles at the vertices of right angles, and acute triangles outside the shape. 3. The four hearts of the regular triangle are one.

The center of gravity is the intersection of the three high lines, the vertical center is the intersection of the perpendicular bisector of each side, and the outer center and the three points form an inscribed circle!

Center of gravity, outer center.

Heart, heart.

The center of gravity is the midline intersection point, and the distance from it to the vertex is twice the distance from it to the midpoint of the opposite edge.

The inner is the intersection of the bisector of the angle (or the center of the circle that is not circled inscribed), and it is at an equal distance from the three sides of the triangle.

The outer center is the middle perpendicular line.

The intersection point (or the center of the circumscribed circle) is an equal distance from the three vertices of the triangle.

The vertical center is the intersection of the high on the three sides of the triangular collapse.

This is called the remnant of the triangle and the four hearts.

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 circle is inscribed in the shape of a mountain horn, which is called the heart of the triangle.

Paracentric: The center of the circumtangent circle of a triangle (the circle tangent to one side of the triangle and the extension lines on the other two sides) is called the paracentricity of the triangle.

Triangle "Five Heart Song".

The triangle has five hearts; Heavy, vertical, internal, external pure or and side heart, the central nature of the five balance pants is very important, carefully grasp the Mo Ji mix

Barycenter. The three middle lines are set to intersect, the position of the intersection is really strange, the intersection is named "center of gravity", the nature of the center of gravity should be clear, the center of gravity divides the middle line segment, and the ratio of several segments is clear;

The ratio of length to length is two to one, and it is good to use it flexibly

Vertical Heart:The triangle is made with three highs, and the three highs must be in the heart

The high line divides the triangle, and there are three pairs of right-angled triangles, and there are twelve right-angled triangles, forming six pairs of similar shapes.

Heart. The triangle corresponds to the three vertices, the corners have bisector lines, and the three lines intersect to determine the common point, which is called the "heart" has roots;

The point to the three sides are equally spaced, which can be made into a triangular inscribed circle, and the center of this circle is called "heart", so it is natural to define it

Circumcenter. The triangle has six elements, and the three inner corners have three sides

Make a perpendicular line on three sides, and the three lines intersect at one point

This point is defined as the "outer center" and can be used as an external circle

The "inside" and "outside" are not confused, and the "inside" and "external" are the key

Draw a picture by yourself according to this, and experience it according to the explanations of others above.

First of all, a triangle has five hearts, not four.

For the triangle's five hearts refer to the center of gravity, the heart, the vertical heart, the outer heart Sun Rang and the side heart, the five of them should not have anything in common!

The vertical center is the intersection of the three high lines, and there is only one.

The heart is the intersection of the three angular bisectors, and there is only one.

The center of gravity is the intersection of the three middle lines, and there is only one.

The outer center is the intersection of the three perpendicular lines, and there is only one.

But the side center is the intersection of the bisector of each corner adjacent to the complementary angle, and there are three.

Heart: The intersection of the bisector of the three angles, which is also the center of the inscribed circle of the triangle.

Properties: Equal distance to three sides.

Outer center: The intersection of the three perpendicular lines, which is also the center of the circumscribed circle of the triangle.

Properties: Equal distances to the three vertices.

Center of gravity: The intersection of the three midlines of early hail.

Properties: The third equinox of the three midlines, the distance to the vertex is 2 times the distance to the midpoint of the opposite side.

Vertical Center: The intersection of the three heights of the straight line.

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

Centricity: The intersection of the outer bisector of any two corners of the triangle and the inner bisector of the third corner.

Properties: Equal distances to three sides.

The four centers of the triangle refer to the center of gravity, the outer center, the inner center, and the vertical center of the triangle. If and only if 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.

1. The center of gravity in mathematics refers to the intersection of the three middle lines of the triangle, and its proof theorems include the dovetail theorem or Seva's theorem, and the application theorems are Menelaus's theorem and Seva's theorem.

2. The intersection of the three high lines of the triangle is called the vertical center of the triangle. The vertical center of an acute triangle is within the triangle; The vertical center of a right-angled triangle is at the right-angled vertex; The perpendicular center of an obtuse triangle is outside the triangle.

3. The intersection of the three inner bisectors of the triangle is called the heart of the triangle. That is, the center of the inscribed circle. The inner is the principle of the intersection of the bisector of the triangle angle: the two tangents of the circle are made at a point outside the circle, and this point is bisected with the line in the center of the circle at the angle between the two tangents.

4. Outer heart is a mathematical term. It refers to the intersection of the perpendicular bisector of the three sides of a triangle, also known as the perpendicular line. Use this point to make the center of the circle to draw the circumscribed circle of the triangle.

The four centers of the triangle refer to the center of gravity, the outer center, the inner center, and the vertical center of the triangle.

Center of gravity theorem: The three midlines of a triangle intersect at a point, and the distance from this point to the vertex is twice the distance from it to the midpoint of the opposite side. This point is called the center of gravity of the triangle.

Centroid theorem: The perpendicular bisector of the three sides of a triangle intersects at a point. This point is called the outer center of the triangle.

Perpendicular theorem: The three highs of a triangle intersect at one point. This point is called the vertical center of the triangle.

Inner theorem: The bisector of the three inside angles of a triangle intersects at one point. This point is called the triangle of the heart.

The center of gravity, the outer center, the vertical center, and the inner center of the triangle are called the four centers of the triangle. They are all important points of relevance for triangles.

The center of gravity, the outer heart, the heart, the heart.

The four hearts in the triangle are:

1. Center of gravity: the intersection of the three middle lines; On the inside of the triangle.

2. Vertical heart. The intersection of the three high days of the wild year; The pure ridge of the acute triangle is vertically centered on the inside, right triangle.

The vertical center is at the right-angled vertice, and the obtuse triangle is on the outside.

3. Heart: the intersection of the bisector of three angles; That is, the center of the inscribed circle of the triangle.

4. Outer center: the vertical bisector of the three sides.

of the intersection. This is the center of the circumscribed circle of this triangle.

The triangle "four hearts" in a plane vector Conclusion:

1. Definition of "Four Hearts":

1) Center of gravity: The intersection of the three sides of the midline, the center of gravity divides the length of the midline into 2:1.

2) Perpendicular: The intersection of three high lines, perpendicular to the corresponding edges.

3) Heart: Regret the intersection of the three bisectors (the center of the inscribed circle), and the distance from any point on the bisector to both sides of the angle is equal.

4) Outer center: The intersection point of the three perpendicular lines (the center of the circumscribed circle), the distance from the center of the outer hail to the vertex of the triangle is equal.

Triangle Quadroid Problem in Planar Vectors:

Vectors are an important concept introduced in high school mathematics and are an important tool for solving geometric problems. In this paper, we summarize the relationship between plane vectors and the four centricities of triangles. In the process of giving conclusions and proving conclusions, the interrelationship between mathematical symmetry and inference can be reflected.

1. Center of gravity (baryce nter).

The center of gravity of the triangle is the intersection of the midlines 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 edge is 2:1. In determining the center of gravity, there is the famous Pappus's theorem.

2. Orthocenter

The intersection of the three high lines of the triangle is called the vertical center of the triangle.

3. Circumcenter

The intersection of the perpendicular bisector (perpendicular) of the three sides of a triangle. Use this point to make the center of the circle to draw the circumscribed circle of the triangle.

Fourth, the heart (incenter).

The intersection of the three inner angles of the triangle is called the heart of the triangle. That is, the center of the inscribed circle.

1. Center of gravity: the intersection of the three middle lines of the triangle. This point must be within the triangle circle, and the point divides each center line into 2:1

2. Heart: the intersection of the three angular lines of the triangle. The point must be within the triangle, and the point must be at equal distances from the three sides, and it is also the center of the inscribed circle of the triangle.

3. Outer center: the intersection of the three perpendicular lines of the triangle. In an acute triangle, the point is inside the triangle, and in an obtuse triangle, the point is outside the triangle.

The distance from the point to the three vertices of the triangle is equal, which is also the center of the circumscribed circle of the triangle.

4. Perpendicular: the intersection of the three heights of the triangular crack. In an acute triangle, the point is inside the triangle, and in an obtuse triangle, the point is outside the triangle.