How to memorize the various hearts of triangles

Center of gravity: The three midlines of the triangle intersect at one point.

Perpendicular: The three high intersections of the triangle at one point.

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

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

Here's a mantra to memorize: heavy middle vertical high, inner angle bisect, outer vertical bisector (I used this formula for three years in high school).

Here's an explanation of the mantra.

Heavy in the middle of the vertical high. Heavy: Center of gravity Middle: Midline.

Vertical: Vertical Height: The height of the triangle.

The inner angle is bisected, and the outer corner is bisected vertically.

Inner: Inner Angular bisection: Angular bisector.

Outside: Outer Center Vertical Bisector: Perpendicular bisector.

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

The inner is the intersection of the three inner bisectors of the triangle, that is, the center of gravity of the inner circle is the intersection of the three middle lines of the triangle.

The outer center is the intersection of the perpendicular bisector of the three sides of the triangle, that is, the side center of the circumscribed circle, which is the intersection of the two outer bisectors and one inner bisector of the triangle In the regular triangle, the center and the center of gravity, the vertical center, the inner center, and the outer center coincide!

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

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.

The outer center is the center of the triangle circumscribed circle, which is the intersection of the perpendicular bisector, and the inner is the center of the inner circumscribed circle, which is the intersection of the angle bisector, and the vertical center can be known from the name as the perpendicular line, that is, the high intersection, and the center of gravity is the intersection of the middle line.

Inside: The intersection of the angular bisector.

Outer center: the intersection of the middle perpendicular line.

Vertical center: high node.

Center of gravity: The intersection of the three midlines The distance from the center of gravity to the vertex is twice the distance from it to the midpoint of the opposite side.

Side: The intersection of an inner bisector and two other vertices of the diplomatic bisector.

1. Center of gravity:Definition: The intersection of three middle lines. Understand: The center of gravity of the segment is the midpoint of the segment. One of the middle lines of a triangle is also a bisector of its area.

2. Inside:Definition: The intersection of three bisectors of the inner angles. The reason is that the points on the bisector of the angle are equally separated from each side of the angle.

3. Outer heart:Definition: The intersection of the three perpendicular bisectors. 【Properties】The distance from the outer center to the vertices of the triangle is equal. The reason for this is that the points on the perpendicular bisector of the segment are at equal distances from the ends of the segment.

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 three middle lines of the center of gravity 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

Make three highs on the vertical triangle, 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 and determine the common point, which is called the "heart" and 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 "inside" and "outside" are not confused, and the "inside" and "external" are the key

The one in April '17 was wrong! It should be the height of the heavy middle vertical, the inner angle bisected, and the outer vertical bisected.

The weight is vertically high, the outer corner is bisected, and the inner vertical is bisected.

Center of gravity: The three midlines of the triangle intersect at a point where 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.

Outer center: The perpendicular bisector of the three sides of the triangle intersects at one point. This point is called the outer center of the triangle.

Perpendicular: The three high intersections of the triangle at one point. This point is called the vertical center of the triangle.

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

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. This point is called the paracentrium of the triangle. The triangle has three paracentricities.

Once you understand the definition, you're fine.

The triangle has five hearts, heavy outside the inner and side hearts, the nature of the five hearts is very important, carefully grasp the three middle lines of the three intersections, the position of the intersection is really kitsch, the intersection is named "center of gravity", the nature of the center of gravity should be clear, the center of gravity is divided into the middle line segment, and the ratio of several segments is clear; The ratio of length to length is two to one, and the triangle has six elements, and the three inner angles have three sides as the middle perpendicular line of the three sides, and the three lines intersect a total of one point

This point is defined as the outer center, and it can be used as an external circle The inner and outer heart are not confused, and the inner and outer circumscribed are the key to make three highs on the triangle, and the three highs must be divided into vertical centers The high line divides the triangle, and there are three pairs of right angles, and there are twelve right triangles, forming six pairs of similar shapes. The 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

Don't confuse the nature of the five hearts, it's really good to start the question.