Mathematically, how many hearts are in a triangle, and what are the hearts of a triangle?

The five hearts of the triangle. One. Theorem.

Center of gravity theorem: The three middle lines of a triangle intersect at a point that reaches the vertex.

The distance is 2 times 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.

Centroid theorem: The bisector of one inner angle of a triangle and the bisector of the outer angle at the other two vertices intersect at one point. This point is called the paracentrium of the triangle. The triangle has three paracentricities.

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

These conclusions had been discovered as early as Euclid's time, and Euclid, with the exception of the vertical theorem, collected them as important theorems in his Geometric Primitives, but later numerous studies on these special points of relevance of triangles and many famous conclusions drawn from them show that the omission of the vertical theorem cannot be regarded as an oversight on the part of the author of the Geometric Originals.

There are 4 commonly used ones.

Outer center: The center of the circumscribed circle is also the intersection of the three perpendicular bisectors of the triangle.

Inner: The center of the inscribed circle is also the intersection of the angular bisector of the three inner angles of the triangle.

Vertical Center: The intersection of three high points.

Center of gravity: The intersection of three midlines (the line segment connecting the midpoint and the diagonal on one side).

The five hearts of the triangle.

The intersection of the bisector of the three inner angles. Inscribed circle center.

The intersection of two outer angles and one bisector of an inner angle. Cut the center of the circle.

The intersection of the three high lines. Orthocenter.

The intersection of the three midlines. Barycenter.

The intersection of the three perpendicular lines. Circumscribed circle center.

The intersection of the high line - the vertical center, the intersection of the middle perpendicular line - the outer center, the intersection of the middle line - the center of gravity, the intersection of the angular bisector - the heart, the intersection of the regular triangle - the center (all), I hope you.

The triangle has an inner center, an outer center, a center of gravity, and a vertical center.

The heart of the heart, the heart of the world.

1. The intersection of the three bisectors of the inner angles of the triangle is called the heart of the triangle. That is, the center of the inscribed circle. The heart is a triangular angular bisector.

The principle of the intersection point: two tangents of the circle are made by a point outside the circle.

This point is bisected by the line at the center of the circle at the angle between the two tangents (proved by congruence).

2. Outer heart is a mathematical term. Refers to the perpendicular bisector of the three sides of a triangle.

Also known as the mid-perpendicular line.

The intersection of . 3. Triangular center of gravity.

is the intersection of the three midlines of the triangle. When the geometric body is a homogeneous object, the center of gravity and centroid.

Superposition. 4. 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; Right-angled triangle.

is centered at a right-angled vertex; Obtuse triangles.

The vertical center is outside the triangle.

5. The side center is the center of the side tangent circle of the triangle (the circle tangent to one side of the triangle and the extension lines on the other two sides).

6. The line connecting the vertex of the triangle and the perimeter midpoint of the opposite side is called the perimeter midline of the triangle. Or the intersection of the three perimeter midlines of a triangle is called the center of the triangle. If a point on one side of the triangle and the vertices on this side split the perimeter of the triangle into two polylines of equal length, then this is called the midpoint of the triangle's perimeter.

The five hearts of the triangle refer to the center of gravity, the outer center, the inner center, the vertical center, and the side center of the triangle. The intersection of the three middle lines is the center of gravity, the intersection of the three vertical bisectors is the outer center, the intersection of the three inner bisectors is the heart, and the intersection of the three high lines of the triangle is the vertical center.

A triangle is a closed figure composed of three line segments in the same plane that are not on the same straight line, which are connected sequentially and have applications in mathematics and architecture. Common triangles are divided into ordinary triangles (the three sides are not equal) and isosceles triangles (isosceles triangles with unequal waist and bottom, and isosceles three-jujube withered angles with equal waist and bottom, that is, equilateral triangles); According to the noisy angle, the stool friend hole is divided into right triangles, acute triangles, obtuse triangles, etc., among which acute triangles and obtuse triangles are collectively referred to as oblique triangles.

Triangle. There are six hearts in the heart: the intersection of the three angular bisectors, which is also the center of the inscribed circle of the triangle.

Properties: Equal distance to three sides.

Circumcenter. Three vertical lines.

The intersection of is also a triangular circumscribed circle.

of the center of the circle. Properties: Equal distances to the three vertices.

Center of gravity: The intersection of the three midlines.

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.

Sexual argument: This point is divided into two parts 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.

Boundary Center: The circumference of the triangle is divided into a 1:1 line and the intersection point of one side of the triangle through the vertex of the triangle.

Properties: The triangle has a total of 3 centricities, and the three centroids are connected to the vertices of the triangle and the three straight lines formed by the stove intersect at one point.

Euler line: The outer center, center of gravity, nine-point circle center, and vertical center of the triangle are located on the same straight line in turn, and this straight line is called the Euler line of the triangle.

Triangle with four centricities. That is, the center of gravity, the heart, the outer heart, and the vertical heart.

Center of gravity: The intersection point of the three middle lines of the triangle, called the center of gravity.

Properties: The ratio of the center of gravity to the two segments of the midline is 2 1.

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

Nature: The distance from the heart to the three sides of the triangle is equal.

Outer center: The intersection of the three perpendicular bisectors. That is, the center of the circle that is excavated outside the triangle.

Properties: The outer branches are spaced at equal distances to the three vertices. Blind.

Vertical center: the intersection of the three highs.

There are three cases: an acute triangle is perpendicular within the triangle, and a right triangle is perpendicular to the right-angled vertice.

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

The triangular silver wisdom shape has two hearts, which are the inner heart and the outer heart. The inner part of a triangle is the center of its inscribed circle, which is the intersection of the bisector of the three inner angles of the triangle. The outer center of a triangle is the center of its circumscribed circle, which is the intersection of the perpendicular bisector of the three sides of the triangle.

The inner part of the triangle is inside the triangle, but the outer heart is likely to be on the outside of it.

Center of gravity: The middle line of the three sides intersects at one point;

Perpendicular: The three heights of the triangle (the straight line) intersect at one point;

Outer center: The perpendicular bisector of the three sides of the triangle intersects at one point;

Inside: The three bisectors of the triangle meet at one point.

The center of gravity of the triangle, the outer center of the mold bridge, the vertical center, the inner center, and the side center are called the five centers of the triangle, and they are all important related points of the triangle.

Side-centered: The bisector of one inner angle of the triangle and the bisector of the outer angle at the other two vertices intersect at one point.

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

2. The center of gravity of the triangle: the intersection point of the three middle lines, 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.

3. The heart of the triangle: the intersection of the three angular bisector lines is the abbreviation of the center of the inscribed circle of the triangle. The distance to the three sides is equal.

4. The outer center of the triangle: the intersection point of the three perpendicular lines is the abbreviation of the center of the circumscribed circle of the triangle. To the three vertices are equal to the distance from the auspicious song.