A closed figure with an internal angle of 180 formed by connecting three sides end to end is called a triangle. Example: △ABC is known, which proves that ∠ ABC+∠ BAC+∠ BCA = 180. Proof: extend BC to point D and cross point C to point E: AB. ∠BAC=∠ACE (two straight lines are parallel and the internal angles are equal) ∠ BCD =180 ∴∠ ACB+∠ ACE+∠ ECD = ∠ BCD = 65438. These two lines are parallel and complementary. The sum of the triangle internal angle and the triangle internal angle is 180 degrees; One outer angle of a triangle is equal to the sum of the other two inner angles; One outer angle of a triangle is larger than the other two inner angles. Proof: It can be proved according to the fact that the sum of the outer angles of a triangle is equal to the inner angles. See How to Prove the Sum of the Internal Angles of a Triangle: Walking into a Triangle (1) for details. Method 1: Tear off three corners of a triangle and put them together to find the sum of the inner corners 180. Method 2: Make an auxiliary line at any vertex of the triangle and find the sum of the internal angles.

Edit this triangle classification.

(1) A. acute triangle: all three angles are less than 90 degrees. It is not a triangle with acute angles, but all three angles are acute angles. For example, an equilateral triangle is also an acute triangle. B. Right triangle (Rt triangle for short): (1) The two acute angles of the right triangle are complementary; (2) The median line on the hypotenuse of the right triangle is equal to half of the hypotenuse; (3) In a right triangle, if there is an acute angle equal to 30, then the right side it faces is equal to half of the hypotenuse. (4) In a right triangle, if a right-angled side is equal to half of the hypotenuse, then the acute angle of this right-angled side is equal to 30 (contrary to (3)); C obtuse triangle: one angle is greater than 90 degrees (acute triangle and obtuse triangle are collectively referred to as oblique triangle). D. Prove congruence separable angle by HL method (2) A. Acute triangle: all three angles are less than 90 degrees. B. right triangle: one angle equals 90 degrees. C. obtuse triangle: one angle is greater than 90 degrees. (acute triangle and obtuse triangle can be collectively called oblique triangle) (3) isosceles triangle with unequal sides; Isosceles triangle (including equilateral triangle).

Edit this paragraph to solve the right triangle:

Pythagorean Theorem only applies to right triangle (called Pythagorean Theorem abroad) A 2+B 2 = C 2, where A and B are two right-angled sides of right triangle and C is hypotenuse. Pythagorean chord number refers to the set of three positive integers that can make Pythagorean theorem relation hold. For example: 3, 4, 5. They are multiples of 3, 4 and 5, respectively. The common numbers of Pythagoras chords are: 3, 4, 5; 6,8, 10; 5, 12, 13; Wait a minute.

Edit the declination triangle in this paragraph.

In triangles A, B and C, if the opposite sides of angles A, B and C are A, B and C respectively, there is a sine theorem (1) a/sina = b/sinb = c/sinc = 2r (circumscribed circle radius is r) (2) cosine theorem. a2 = B2+C2-2bc * COSAB 2 = a2+C2-2ac * COSBC 2 = a2+B2-2ab * COSC(3)Cosa =(B2+C2-)

Edit the properties of this triangle.

1. The sum of any two sides of the triangle must be greater than the third side, which also proves that the difference between any two sides of the triangle must be less than the third side. 2. The sum of the internal angles of the triangle is equal to 180 degrees. 3. The bisector of the vertex, the midline of the bottom and the height of the bottom of the isosceles triangle coincide, that is, the three lines are one. 4. The square sum of two right angles of a right triangle is equal to the square-pythagorean theorem of the hypotenuse. The center line of the hypotenuse of a right triangle is equal to half of the hypotenuse. 5. The outer angle of a triangle (the angle formed by one side of the inner angle of the triangle and the extension line of the other side) is equal to the sum of two non-adjacent inner angles. 6. A triangle has at least two acute angles. 7. Angle bisector of triangle: the bisector of an angle of triangle intersects with the opposite side of this angle, and the line segment between the vertex and the intersection of this angle. 8. In an isosceles triangle, the bisector of the vertex of the isosceles triangle bisects the base and is perpendicular to the base. 9. Pythagorean inverse theorem: If three sides of a triangle have the following relationship (A 2+B 2 = C 2. ) Then this triangle must be a right triangle. 10. The sum of the outer angles of a triangle is 360 degrees. 1 1. The base and the triangle with the same height have the same area. 12. The area ratio of equilateral triangles is equal to their height ratio, and the area ratio of equilateral triangles is equal to their base ratio. 13. The sum of squares of the lengths of the three center lines of a triangle is equal to 3/4 of the sum of squares of the lengths of its three sides. 14. tanatantbank = tana+tanb+tanc is always satisfied in △ABC. 15. The outer angle of a triangle is larger than any inner angle that is not adjacent to it. 16. The corresponding edges of congruent triangles are equal, and the corresponding angles are equal. 17. The center of the triangle is at the intersection of the three midlines. At least one angle in the 18 triangle is greater than or equal to 60 degrees, and at least one angle is less than or equal to 60 degrees.

Edit the five centers, four circles, three points and a line of this triangle.

Five hearts, four circles, three points and one line of a triangle.

These are the special points of the triangle and the related geometric figures based on these special points. "Five minds" refers to external mind, vertical mind, internal mind, external mind and cross heart; "Four circles" are inscribed circle, circumscribed circle, circumscribed circle and Euler circle; "Three points" are Lemmon point, nagel point and Euler point; The "first line" is the Euler line. Let's note that the three vertices of a triangle are A, B and C, and the corresponding opposite lengths are A, B and C, and the coefficient k (A) =-A 2+B 2+C 2, K(b) and K(c) and so on. Each component of the three-line coordinate can be directly multiplied by the corresponding side length and converted into the area coordinate. The method of calculating the plane rectangular coordinates of a point by combining the area coordinates of a point with the coordinates of three vertices is as follows: the area coordinates of a point are (μa, μb, μc), and the sum of the three components is μ, so there is px = (μ a xa+μ b xb+μ c xc)/μ, Py and so on. Five-center names define three-line coordinates (internal coordinates) and area coordinates (barycenter coordinates).

The intersection of the three center lines of the center of gravity (the connecting line from the vertex to the midpoint of the opposite side) is1/a:1/b:1/c1.

Intersection of three heights (opposite sides perpendicular to the vertex) at the vertical center SECA: SECB: SECC1/K (a):1/K (b):1/K (c) or tan(A): tan(B): tan(C).

Intersection point of three bisectors of the interior angle1:1:1a: b: c.

The epicentre COSA: COSB: Intersection point of three vertical lines of COSCA 2 k (a): B 2 k (b): C 2 k (c)

The intersection of the bisector of one side center and the bisectors of the other two angles is-1: 1: 1, and so on. A: B: C, etc.

Four-circle inscribed circle: a circle with the center as the center and the distance from the center to the edge as the radius, which is tangent to all three sides of a triangle. Circumcircle: A circle with the radius of the center and the distance from the center to the vertex. All three vertices of a triangle are on the circumference. Secant circle: a circle with the center as the center and the distance from the center to the edge as the radius, which is tangent to the extension line of one side and the other two sides of a triangle. Euler circle: also known as "nine-point circle", that is, three Euler points, three midpoint, three high vertical nine-point circle. The center of the nine-point circle is the midpoint of the connecting line between the vertical center and the epicenter. The three-line coordinates are: cos(B-C): cos(C-A): cos(A-B), and the radius is half of the radius of the circumscribed circle. The inscribed circle is tangent to the Euler circle at a certain Euler point. Three-point names define three-line coordinates.

The intersection of the three vertices of the Lemmon point and the tangent point of the inscribed circle is also called the similar center of gravity A: B: C.

The intersection of the three vertices of Nagel point and the tangent point of the tangent circle is also called the boundary center CSC 2 (a/2): CSC 2 (b/2): CSC 2 (c/2).

The midpoint of the line connecting the three vertices of Euler's point to the vertical center is also called Feuerbach point (temporarily vacant).

A four-point line with a vertical center, a center of gravity, an outer center and a center at nine o'clock is called Euler line.

Why is the triangle in this section stable?

If two sides of a triangle are selected, the non-common endpoints of the two sides are connected by the third side; The third side cannot be stretched or bent; The distance between the two ends is fixed; The included angle between the two sides is arbitrary; All three angles of a triangle are fixed, and then the triangle is fixed; Triangle stability; Select any n-sided shape. Then the non-common endpoints on both sides are connected by more than one edge ∴ The distance between the two ends is not fixed ∴ The included angle between the two sides is not fixed ∴n-polygon (n≥4) is not fixed at each angle, so the N-polygon (n≥4) is unstable.

Edit the relationship between the angles of the triangle in this paragraph.

(1) The sum of the three internal angles of the triangle is equal to 180 (on the spherical surface, the sum of the internal angles of the triangle is greater than180); (2) An outer angle of a triangle is equal to the sum of two inner angles that are not adjacent to it; (3) The outer angle of a triangle is larger than any inner angle that is not adjacent to it; (4) The sum of two sides of the triangle is greater than the third side, and the difference between the two sides is less than the third side; (5) In the same triangle, the big side is opposite to the big corner, and the big corner is opposite to the big side. (6) There are four special segments in the triangle: angle bisector, midline, height and midline. (Note ①: In an isosceles triangle, the bisector of the top angle, the middle line and the high line overlap with each other ②: The middle line of the triangle is the connecting line between the two sides. It is parallel to the third side and equal to half of the third side. (7) The intersection point of the bisector of a triangle is called the center of the triangle, which is the center of the inscribed circle of the triangle, and its distance to each side is equal. (8) The center of the circumscribed circle of a triangle, that is, the outer center, is the intersection of the perpendicular lines of the three sides of the triangle, and its distances to the three vertices are equal. (9) The intersection of the three center lines of a triangle is called the center of gravity of the triangle, and its distance to each vertex is equal. The intersection of three heights of a (10) triangle is called the vertical center of the triangle. The median line of the triangle (1 1) is parallel to the third side and equal to 1/2 of the third side. (12) The included angle between one side of a triangle and the extension line on the other side is called the outer angle of the triangle. Note: ① The heart and center of gravity of the triangle are all inside the triangle. ② The obtuse triangle is vertical, and the outer center is outside the triangle. ③ A right triangle has a vertical center and an outer center on the side of the triangle. The vertical center of a right triangle is the right vertex and the outer center is the midpoint of the hypotenuse. (4) Both the vertical center and the outer center of the acute triangle are inside the triangle. The sum of two sides of a triangle is greater than the third side, and the difference between the two sides is less than the third side. One application! The maximum area of a triangle with a fixed perimeter