What are the intersection of the bisector of the triangle angle, the intersection of the perpendicul

The intersection of the angular bisector is called the heart.

The intersection of the perpendicular lines is called the vertical letter.

The intersection points of the midline are called the center of gravity.

The intersection of the perpendicular bisector is called the outer center.

The triangle intersects one of the inner bisectors and the outer bisectors at the other two vertices. It's called the side heart. (Note that it is the outer corner).

Have fun.

Heart, heart, center of gravity.

My method of memorization: the distance between the bisector of the angle and the two sides of the angle is equal, so it can be made into a triangle of the inscribed circle, that is, the heart;

The distance from the middle perpendicular line to both sides of the line segment is the same, so it can be made as a triangular circumscribed circle, that is, the outer center;

The perpendicular line is the vertical center; The midline is the center of gravity, and the middle and accent sounds are similar.

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. Exocentric theorem:

The perpendicular bisector of the three sides of the triangle intersects at one point. Perpendicular theorem: The three highs of a triangle intersect at one point.

Inner theorem: The bisector of the three inside angles of a triangle intersects at one point.

The intersection of the bisector of the triangle angle is called the heart, the intersection of the perpendicular line (that is, the height) is called the vertical center, and the intersection of the middle line is called the center of gravity. There is also the intersection of the perpendicular bisector called the outer center.

The three middle lines intersect at one point, the center of gravity.

The perpendicular bisector of the three sides intersects at one point. Circumcenter.

Three of a kind is high at one point. Orthocenter.

The three inner bisectors intersect at one point. Heart.

One of the inner bisectors and the outer bisectors at the other two vertices intersect at one point. Side-by-side.

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 inner is the principle of the intersection of the bisector of the triangle angle: the two tangents of the circle are made by the outer point of the circle, and this point is bisected with the line of the center of the circle at the angle between the two tangents (proved by congruence).

Triangle with five hearts

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 circle tangent to one side of the triangle and the extension lines on both sides is called a circumscient circle of the triangle, and the center of the circumscribed circle is called a triangular paracentric.

The circle tangent to each side of the triangle is called the inscribed circle of the triangle, and the center of the inscribed circle is the inner circle of the triangle, and the distance between the inner circle and the three sides of the triangle is equal. This triangle is called the circumscribed triangle of the circle. A triangle has one and only one inscribed circle.

The center of gravity of the triangle is the intersection of the midlines of the three sides of the triangle;

The vertical center is the intersection of the high lines on the three edges;

The outer center is also the center of the circumscribed circle of the triangle, that is, the intersection of the perpendicular bisector of the three sides of the triangle;

The inner is the center of the inscribed circle, which is the intersection of the three bisectors of the triangle.

Characteristics of the intersection of the angular bisector of the triangle: the three angular bisectors of the triangle intersect at one point, which is called the center of the triangle inscribed circle.

The theorem of the nature of the bisector of the inner angle of the triangle: the bisector of the inner angle of the triangle is divided into two line segments on opposite sides, then these two line segments are proportional to the correspondence of the two sides of the angle.

The determination theorem of the bisector of the inner angle of a triangle: In ABC, if the point D divides the side BC according to the ratio of the edge AB and the side AC, then the line segment AD is the bisector of the bac.

The intersection of the three angular bisectors of a triangle is equal to the distance from the three sides.

The inner heart is the intersection of the three bisectors of the inner angles of the triangle, that is, the center of the inscribed circle.

The inner is the principle of the intersection of the angular bisector of the triangle: the meridian is made into two tangents of the circle at a point outside the circle, and this point is bisected with the line at the center of the circle to the angle between the two tangents (principle: the distance from the point to both sides of the angle on the angle bisector is equal).

Inner theorem: The angular bisector of the three inner angles of a triangle intersects at a point. This point is called the triangle of the heart.

Noting that the distance from the heart to the three sides is equal (the radius of the inscribed circle), the heart theorem is actually very easy to prove.

If the three sides are L1, L2, L3, and the circumference is P, then the center of gravity coordinates of the heart are (L1 P, L2 P, L3 P).

The distance from the inside of a right triangle to the side is equal to one-half of the difference between the two right sides and minus the hypotenuse.

The projection of the inner part of the triangle formed by the upper point and the two foci on the hyperbola on the real axis is the vertex of the corresponding branch.

1. The bisector of the three inner angles of the triangle intersects at a point, which is called the heart of the triangle. The distance from the heart to the three sides is equal. This is because the heart is the circle of the triangular inscribed circle, and the distance from the heart to the three sides is equivalent to the radius of the inscribed circle, which is naturally equal.

2. Right triangle.

The distance from the heart to the three sides is equal to one-half of the difference between the two right-angled sides and minus the hypotenuse.

3. Hyperbola.

On any one of the upper points and two foci of a triangle. The projection of the heart on the real axis is the apex of the corresponding branch.

4. The inner is the principle of the intersection of the triangular angle bisector trapped in history: two tangents of the circle at a point outside the circle.

This point is equal to the line at the center of the circle, which divides the angle between the two tangent lines.

The intersection of the angular bisector is called a triangleHeart. , the intersection point of the vertical lines, which is the intersection point of the high line, is calledOrthocenter.

Triangle Inside: A triangle inside is when the bisector of the three corners of the three inner corners intersect at a point, and this point is called the triangle inside. This point is also the center of the inscribed circle of this triangle. The distance from the inside of the triangle to the three sides of the triangle is equal.

Triangle Vertical Center: The vertical center of a triangle refers to the intersection of the three heights of the triangle or its extension line at one point, which is called the vertical center of the triangle. The vertical center of an acute triangle is inside the triangle, the vertical center of a right triangle is on the vertex of the right angle, and the vertical center of an obtuse triangle is outside the triangle.

Triangles have many properties, and there are many "mind" properties:

1. Center of gravity: The center of gravity of the triangle is the intersection of the three middle lines of the triangle. When the geometry is a homogeneous object, the center of gravity coincides with the centroid.

2. Outer center: The center of the triangle circumscribed circle is called the outer center of the triangle. The center of the circumscribed circle of the triangle is the intersection of the perpendicular bisector of the three sides of the triangle, and the three vertices of the triangle are on this circumscribed circle.

3. Side center: the center of the triangle side tangent circle, referred to as the triangle side center, which is the intersection of the bisector of one inner angle of the triangle and the outer angle bisector of the two inner angles of the front Senta; Obviously, any triangle has three side-cut circles and three side-centers.

Note: In an equilateral triangle, the center of gravity, the vertical center, the inner center, the outer center, and the paralateral center are all one, i.e., the five "hearts" are all located at the same point in the equilateral triangle.

It's called the heart. The three bisectors of the triangle intersect at a point where the inside of the triangle is the heart of the triangle. This point is also the center of the inscribed circle of the triangle.

The distance from the inside of the triangle to the three sides of the triangle is equal. Correspondingly, there is the outer center of the triangle, and the center of the triangle is called the outer center of the triangle. The center of the circumscribed circle of the triangle is the intersection of the perpendicular bisector of the three sides of the triangle, and the three vertices of the triangle are on this circumscribed circle.

1.The three bisectors of the triangle intersect at a point where the inside of the triangle is the heart of the triangle.

2.The distance from the inside of a right triangle to the side is equal to the difference between the two right angles and minus the hypotenuse.

3.The distance from the heart to the three sides of the triangular infiltration forest shape is equal, and they are all equal to the radius r of the inscribed circle.

is the heart of the triangle, a, b, c are the three vertices of the triangle, extending ao to cross bc side to n, then ao:on=ab:bn=ac:cn=(ab+ac):bc.

5.Euler's theorem: In a triangle, if r and r are the radii of the circumscribed circle and the inscribed circle, respectively, and o and i are its outer and inner centers, respectively, then oi = r -2rr.