What are the five hearts of a triangle?

1, center of gravity: the intersection of three midlines of a triangle.

2. Exterior center: the perpendicular bisector intersection of three sides of a triangle.

3. Hanging down: The three heights of the triangle intersect at one point.

4. Heart-shaped: The bisectors of the three internal angles of a triangle intersect at one point.

5. Center: Only when the triangle is a regular triangle, the center of gravity, center of gravity, inner heart and outer heart are connected into a heart, which is called the center of a regular triangle.

Five-center feature of triangle;

1, heart: the intersection of bisectors of three internal angles of a triangle, that is, the center of the inscribed circle. The principle that the center is the intersection of the bisector of a triangle: make two tangents of the circle through a point outside the circle, and the connecting line between this point and the center of the circle bisects the included angle of these two tangents (principle: the distance from the point on the bisector of the angle is equal to both sides of the angle).

2. Outer center: it is the intersection of the perpendicular lines of the three sides of the triangle, that is, the center of the circumscribed circle. 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.

3. Center: A triangle has only five centers of gravity, namely, vertical center, inner center, outer center and transverse center. If and only if the triangle is a regular triangle, the four centers are combined into one, which is called the center of the regular triangle.

4. Center of gravity: The center of gravity is the intersection of the center lines of the triangle.

5.paracenter: The point where a bisector of the inner corner of a triangle intersects with the bisectors of the outer corners of the other two corners is the Paracenter of the triangle. The distance from the center of the side of a triangle to the three sides is equal. A triangle has three tangent circles and three edge centers. The center of mass must be outside the triangle. The radius of the tangent circle on the hypotenuse of a right triangle is equal to half the circumference of the triangle.

Extended data:

Any triangle has five hearts, namely, center of gravity, center of gravity, outer center, inner center and edge center.

Center of gravity: the intersection of three sides of a triangle is the center of gravity of the triangle; Inside the triangle;

Gravity center theorem: the distance from the gravity center to the vertex is twice the distance from the opposite midpoint.

Vertical center: the intersection of three high lines of a triangle is the vertical center of the triangle; The vertical center of the acute triangle is included, the right triangle is at the right vertex, and the obtuse triangle is outside.

Exterior center: the intersection of the perpendicular lines of the three sides of a triangle is the exterior center of the triangle; The outer center of acute triangle is inside, right triangle is at the midpoint of hypotenuse, and obtuse triangle is outside; This point is the center of the △ circumscribed circle, and the distances to the three vertices are equal. This distance is called the radius r of the circumscribed circle.

Inner heart: the intersection of bisectors of three internal angles of a triangle is the inner heart of the triangle; Inside the triangle, this point is the center of the inscribed circle of the triangle, and the distance to the three sides is equal. This distance is the radius of the inscribed circle R.

References:

Baidu encyclopedia-triangle center