The nature of the four important segments of a triangle

Center of gravity: The line connecting the vertices of the triangle and the midpoint of the opposite side intersect at one point, which is called the center of gravity of the triangle;

Perpendicular: The high intersection of each side of the triangle at one point is called the vertical center of the triangle;

Outer centric: The vertical bisector on each side of the triangle intersects at a point, which is called the outer center of the triangle;

Heart: The bisector of the three inner angles of the triangle intersects at one point, which is called the triangle heart;

Center: The center of gravity, vertical center, outer center, and inner center of a regular triangle coincide, which is called the center of the regular triangle.

Triangle "Five Heart Song".

The triangle has five hearts; Heavy, vertical, inner, outer and side minds, the nature of the five hearts is very important, and it is important to carefully grasp it

The center of gravity three middle lines are set to intersect, the intersection position is really kitsch, 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

The vertical center triangle is made of three highs, and the three highs must be in the vertical 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.

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

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

The outer triangle has six elements, and the three inner angles 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 "inner heart" and "outer heart" should not be confused, and the "internal incarnation" and "external connection" are the key side hearts. The center of a triangle's atangent circle (a circle tangent to one side of the triangle and the extension lines on the other two sides) is called a paracentric. The centroid is the intersection of the bisector of the inner angle of a triangle and its two outer bisectors that are not adjacent to each other, and it is at an equal distance from the three sides of the triangle.

As shown in the figure, the point M is a side of ABC. The intersection of the outer bisector of any two corners of the triangle and the inner bisector of the third corner. A triangle has three paracentricities, and it must be outside the triangle.

If o is the paracenter of abc, expressed by a vector, then there is aoa=bob=coc1, 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, and the point is the paracenter of the triangle.

2. Each triangle has three side centers.

3. The distance from the side center to the three sides is equal.

Middle line: The triangle can be divided into two triangles of the same area;

Angle bisector: divides one of the top angles of a triangle into two identical parts;

High line: is the perpendicular line from the edge to the corresponding point, and the shortest distance segment.

The properties of the midline of the triangle are:

1. The three middle lines of the triangle are all within the Bisui Tsai triangle.

2. The three middle lines of the triangle intersect at one point, which is called the center of gravity of the triangle.

3. Right triangle.

The middle line on the hypotenuse is equal to 1 2 on the hypotenuse.

4. The area of the triangle formed by the family of the triangle midline.

It is equal to 3 4 of the area of this triangle5. Triangular center of gravity.

The middle line is divided into two line segments with a length ratio of 1:2.

The center line is the segment of a triangle that connects from the midpoint of a side to a diagonal vertex.

The three midlines of a triangle always intersect at the same point, and this point is called the center of gravity of the triangle, which is divided into 2:1 (vertex to centr: center of gravity to the midpoint of the opposite side).

Nature of the midline:

1. The three middle lines of any triangle divide the triangle into six parts of equal area. The middle line divides the triangle into two parts of equal area. Other than that, any other straight line passing through the midpoint does not divide the triangle into two parts of equal area.

2. The intersection point of the middle line of the triangle is the center of gravity, and the center of gravity is 2:1 (vertex to center of gravity: center of gravity to the midpoint of the opposite side).

3. In a right-angled triangle, the middle line on the side corresponding to the right angle is half of the hypotenuse.

The use of the midline:

1. The function of the center line is to ensure the voltage of each phase when the load is asymmetrical.

It's still symmetrical and all work fine; If one phase is disconnected, it will only affect the load of the original phase, and not the other two phases.

2. The three middle lines of any triangle divide the triangle into six parts of equal area.

3. It is bisecting the opposite side, and the triangle can also be divided into two parts of equal area, which is used to verify the congruent triangle and the middle line of the triangle.

It is a line segment that connects a vertex and the opposite side of a triangle, and a triangle has 3 center lines.

Lines related to the triangle auspicious Zen: bisectoral line, perpendicular line, middle line.

1. Angular bisector.

The bisector of an inner corner of a triangle intersects its opposite edge, and the segment between the vertices and intersections of this corner is called the angular bisector of the triangle.

The angular bisector theorem of a triangle: The distance from the point to the two sides of the angle on the angular bisector of a triangle is equal.

2. Perpendicular.

A perpendicular segment from the vertex of a triangle to its opposite side or to the extension of the opposite side is called the height (also called the perpendicular line) on the edge of the pair.

3. The middle line. The line between the vertices of a triangle and the midpoint of its opposite edge is called the midline of that opposite edge.

A triangle is a shape consisting of three line segments that are not on the same line that meet each other end in order. The angle formed by two adjacent sides is called the inner angle of the triangle, which is referred to as the angle of the triangle.

A triangle is made up of 3 line segments. A line segment means that there are endpoints at both ends, which cannot be extended, which is different from straight lines and rays.

A triangle is a closed plane shape composed of three line segments connected end to end, which is the most basic polygon. A closed shape consisting of three line segments that are not on the same line is connected one after the other, called a triangle. The figure enclosed by three straight lines on the plane or three arcs on the sphere, and the figure enclosed by the three straight lines is called a plane triangle;

The shape enclosed by three arcs is called a spherical triangle, also known as a trilateral. The closed geometry obtained from three line segments connected end to end is called a triangle. A triangle is the basic shape of a geometric pattern.

The important line segments of the triangle are high, midline, angular bisector, etc.

1. High. Definition of the height of the triangle: from one vertex of the triangle to the opposite side of the triangle to make a perpendicular line where the mausoleum is located, the line segment between the vertex and the perpendicular foot is called the height on the side of the triangle, referred to as the height of the triangle.

Note: Height is different from perpendicular lines, where high is a line segment and perpendicular lines are straight lines.

1) The triangle is three high; (2) The intersection of the three heights of the triangle is called the vertical center of the triangle; (3) The three heights of an acute triangle are inside the triangle, one of the right triangles is high inside the triangle, the other two are exactly its two sides, and the obtuse triangle is one high inside the triangle and the other two are outside the triangle.

Intersection of the high triangles: The intersection of the three heights of an acute triangle is on the inside of the triangle, the intersection of the three heights of a right triangle is at the vertex of the right angle, and the intersection of the three heights of an obtuse triangle is on the outside of the triangle.

2. The middle line. Definition of the center line of the triangle: In a triangle, the line segment that connects a vertex to the midpoint of its opposite side is called the center line on this side of the triangle.

1) Any midline of the triangle will divide the triangle into two triangles of equal area; (2) The triangle has three middle lines, all of which are inside the triangle and intersect at one point, which is called the center of gravity. (3) The distance from the center of gravity to a vertex is twice the distance from the center of gravity to the midpoint of the opposite side of the vertex.

3. Angular bisector.

Definition of the angular bisector of a triangle: In a triangle, the bisector of an inner angle intersects its opposite side, and the line segment between the vertices and the intersection of this angle is called the angular bisector of the triangle.

1) The distance from the point on the angular bisector of the triangle to the two sides of the triangle on both sides of the angular bisector is equal; (2) The triangle has three angular bisectors, all of which are inside the triangle; (3) The bisector of the three corners of the triangle intersects at one point, and the distance from this point to the three sides of the triangle is equal, which is the center of the inscribed circle of the triangle and becomes the heart of the triangle.

There are three important line segments in a triangle: the angular bisector, the midline, and the height.

1.Angle bisector: The bisector of an inner angle of a triangle intersects the opposite side of the angle, and the line segment between the vertices and intersections of this angle is called the angular bisector of the triangle.

2.Midline: In a triangle, the segment that connects a vertex to its midpoint on the opposite side is called the midline of the triangle.

3.Height: A perpendicular line from one of the vertices of the triangle to the line where its opposite edge is located, and the line segment between the vertex and the perpendicular foot is called the height of the triangle.

Instructions:1The angular bisector line, the middle line, and the high overlap section of the triangle are all line segments;

2.The angular bisector and the middle line of the triangle are both inside the triangle and intersect at one point; The height of a triangle may be inside the triangle (acute triangle), outside (obtuse triangle), or on the edge (right triangle), and they (or extensions) intersect at a point.

Three line segments of the triangle. Common triangles are divided into ordinary triangles (the three sides are not equal), isosceles triangles (isosceles triangles with unequal waist and bottom, and isosceles triangles with equal waist and bottom, that is, equilateral triangles); According to the angle, there are right-angled triangles, acute triangles, obtuse triangles, etc.

A closed figure obtained from the end of three line segments that are not in the same line in the same and sell planes. The sum of the three interior angles of a triangle is equal to 180 degrees. The sum of any two sides of the triangle is greater than the third side.

The difference between the two sides of the triangle is less than that of the third side. The outer angles of a triangle are equal to the sum of the two inner angles that are not adjacent to it.