Vertical center: the intersection of high lines.
Outer center: the intersection of the middle perpendicular of three sides and the center of the circumscribed circle.
Inner heart: the intersection of the bisector of the triangle angle and the center of the inscribed circle.
The five centers of a triangle
A theorem
Gravity center theorem: Three midlines of a triangle intersect at a point, which is the vertex.
The distance is twice the distance from the opposite midpoint. This point is called the center of gravity of the triangle.
Eccentricity theorem: the perpendicular lines of three sides of a triangle intersect at one point. This point is called the outer center of the triangle.
Vertical center theorem: three heights of a triangle intersect at one point. This point is called the center of the triangle.
Interior Theorem: The bisectors of three interior angles of a triangle intersect at one point. This point is called the heart of a triangle.
Proximity theorem: the bisector of the inner angle of a triangle intersects the bisector of the outer angle of the other two vertices at one point. This point is called the center of the triangle. A triangle has three side centers.
The center of gravity, outer center, vertical center, inner center and lateral center of a triangle are called the five centers of the triangle. Are all important connection points of the triangle.
The above conclusion was discovered as early as Euclid's time. In addition to sag theorems, Euclid collected them as important theorems in his Elements of Geometry. However, many studies on these special related points of triangles and many famous conclusions drawn from them later show that the omission of sag theorem is the negligence of the author of Geometry Elements.
Supplementary song formula:
A triangle has five hearts; Heavy-duty, vertical, internal, external and lateral centers,
The essence of five minds is very important, and it is very important to master it carefully.
Chongxin
The three midlines must intersect, and the intersection position is really strange.
The intersection is named "center of gravity", and the nature of the center of gravity should be clear.
In the center of gravity segmentation, you can hear the ratio of line segment to several segments;
The ratio of length to length is two to one, so we should use it flexibly and master it well.
worry (about/over)
If a triangle is three high points, then the three high points must cross.
The high line is divided into triangles with three pairs of right angles.
There are twelve right triangles, forming six pairs of similar shapes.
There are four points in the picture, which can be found clearly after careful analysis.
heart
A triangle corresponds to three vertices, and each corner has a bisector.
The intersection of three lines has a * * * point, which is called "inner heart" and has roots;
The points to the three sides are equidistant and can be used as the inscribed circle of a triangle.
This round heart is called "inner heart", so it is natural to define it.
External heart
A triangle has six elements, three inner corners and three sides.
Make the middle of the three sides vertical and the three lines intersect at one point.
This point is defined as the "outer center" and can be used as the circumscribed circle.
Don't confuse "inner heart" with "outer heart". "Inner cut" and "outer connection" are the key.
Draw a picture of yourself according to this and compare it with other people's explanations above.
The center of gravity is the intersection of the center line and the center is the intersection of the bisector (or the center of the inscribed circle).
The epicenter is the intersection of the middle vertical lines (or the center of the circumscribed circle), and the vertical center is the intersection of the high lines.
These are called the four centers of a triangle.
There is also a center called Paracenter: the intersection of the bisectors of the outer corners (there are three), (or the center of the tangent circle).
Only a regular triangle has a center. At this time, the center of gravity, inner heart, outer heart, hanging heart and four hearts are integrated.
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