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. A triangle has six centers: inner center, outer center, center of gravity, vertical center and Euler line.

Heart: The intersection of bisectors of three angles is also the center of the inscribed circle of a triangle.

Attribute: The distances to three sides are equal.

Eccentricity: the intersection of three perpendicular lines is also the center of the circumscribed circle of the triangle.

Attribute: The distances to the three vertices are equal.

Center of gravity: the intersection of three midlines.

Property: The distance from the bisector of the three median lines to the vertex is twice the distance from the midpoint of the opposite side.

Vertical center: the intersection of straight lines of three heights.

Attribute: This point is divided into two parts of each high line.

Paracenter: the intersection of the bisector of the outer corner of any two angles of a triangle and the bisector of the inner corner of the third angle.

Attribute: The distances to three sides are equal.

Centroid: Through a vertex of a triangle, divide the perimeter of the triangle into the intersection of 1: 1 and a straight line of one side of the triangle.

Properties: A triangle * * * has three boundary centers, and three straight lines connecting these three boundary centers and their corresponding triangle vertices intersect at one point.

Euler Line: The outer center, center of gravity, center of nine points and vertical center of a triangle are located on the same straight line in turn. This straight line is called the Euler line of triangle.

6. 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 the inner angles that are not adjacent to it.

7. A triangle has at least two acute angles.

8. 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 point of this angle is called the angle bisector of triangle.

9. In an isosceles triangle, the bisector of the vertex of the isosceles triangle bisects the base and is perpendicular to the base.

10. Pythagorean inverse theorem: If three sides of a triangle have the following relationship, then a? 0? 5+b？ 0? 5=c？ 0? five

Then this triangle must be a right triangle.

The relationship between the angles of a triangle

The sum of (1) triangles is equal to180;

(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 faces the big corner, and the big corner faces the big side.

(6) Four special line segments in the triangle: angle bisector, midline, height and midline.

(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 midlines of a triangle is called the center of gravity of the triangle, and its distance to each vertex is equal to twice its distance to the midpoint of the opposite side.

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.

Note: ① The heart and center of gravity of the triangle are all inside the triangle.

② An obtuse triangle has its vertical center and its 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.

special triangle

1. similar triangles

(1) Two triangles with the same shape but different sizes are called similar triangles.

(2) The nature of similar triangles

The corresponding sides of similar triangles are proportional and the corresponding angles are equal.

The ratio of the corresponding sides of similar triangles is called similarity ratio.

The perimeter ratio of similar triangles is equal to the similarity ratio, and the area ratio is equal to the square of the similarity ratio.

Similar triangles's corresponding line segments (angular bisector, median line and height) are equal.

(3) similar triangles's judgment

1 If three sides are proportional, two triangles are similar.

Two triangles are similar. If the two sides are proportional, the included angle is equal.

Two triangles are similar if the corresponding angles are equal.

2. congruent triangles

(1) Two triangles that can completely coincide are called congruent triangles.

(2) The nature of congruent triangles.

The corresponding angles (edges) of congruent triangles are equal.

Congruent triangles's corresponding line segments (angle bisector, median line and height) are equal, with equal perimeter and equal area.

(3) congruent triangles's judgment

① sas2asa ③ aas4sss5hl (RT triangle)

3. isosceles triangle

The nature of isosceles triangle;

(1) The two base angles are equal;

(2) The bisector of the vertex angle, the median line on the bottom edge and the height on the bottom edge coincide with each other;

Determination of isosceles triangle;

(1) equilateral;

(2) The two base angles are equal;

4. equilateral triangle

Properties of equilateral triangle:

(1) The bisector of the top corner, the median line on the bottom edge and the height on the bottom edge coincide;

(2) All angles of an equilateral triangle are equal and equal to 60.

Determination of equilateral triangle;

(1) A triangle with three equal angles is an equilateral triangle;

(2) An isosceles triangle with an angle equal to 60 is an equilateral triangle.

Area formula of triangle

(1)S△= 1/2*ah(a is the base of the triangle, and H is the height corresponding to the base).

(2) s △ =1/2 * AC * sinb =1/2 * BC * Sina =1/2 * ab * sinc (the three angles are ∠A∠B∠C, respectively, and the opposite side

(3)S△=√S *(S-a)*(S-b)*(S-c)S = 1/2(a+b+c)

(4)S△=abc/(4R)R is the radius of the circumscribed circle.

(5)S△= 1/2*(a+b+c)*r r is the radius of the inscribed circle.

(6) | a b 1 |

S△= 1/2 * | c d 1 |

| e f 1 |

| a b 1 |

| c d 1 | is a third-order determinant, and this triangle ABC is in the plane rectangular coordinate system A(a, b), B(c, d), C(e, f), where ABC.

| e f 1 |

It is best to choose from the upper right corner in counterclockwise order, because the results obtained in this way are generally positive. If you don't take this rule, you may get a negative value, but it doesn't matter, just take the absolute value and it won't affect the size of the triangle area!

Triangular objects in life

Umbrellas, hats, colorful flags, lampshades, sails, pavilions, snow-capped mountains, roofs, watermelons cut into triangles, torch ice cream, edge lines of tropical fish, butterfly wings, rockets, bamboo shoots, pagodas, pyramids, briefs, triangle iron for machines, some road signs, the Yangtze River Delta, cable-stayed bridges, etc.

Conditions for congruence of triangles Note: Only three angles are equal, so two congruences of triangles cannot be deduced.

(1) Two triangles with equal sides are equal, abbreviated as "SSS".

(2) Two triangles whose two corners and their sides are congruent, abbreviated as "ASA".

(3) The opposite side of two angles and one angle corresponds to the congruence of two triangles, abbreviated as "AAS".

(4) Two triangles with equal included angles are congruent, abbreviated as "SAS".

(5) The hypotenuse and a right-angled side correspond to the congruence of two right-angled triangles, abbreviated as "HL".

The nature of congruent triangles

The corresponding angles of congruent triangles are equal, and the corresponding edges are equal.

Line segment in triangle

Midline: the line connecting the vertex and the midpoint of the opposite side, bisecting the triangle.

Height: the line from the vertex to the opposite vertical foot.

Angular bisector: a straight line from a vertex to a point with equal distance on both sides.

Midline: a line connecting the midpoints of any two sides.

Triangle correlation theorem

Gravity center theorem

The three median lines of a triangle intersect at a point, and the distance from the point to the vertex is twice as long as the distance from the midpoint of the opposite side to the vertex.

The intersection above is called the center of gravity of the triangle.

Eccentricity theorem

The perpendicular bisector of three sides of a triangle intersect at one point.

This is called the outer center of a triangle.

Vertical center theorem

The three heights of a triangle intersect at a point.

This is called the center of the triangle.

Internal theorem

The bisectors of the three internal angles of a triangle intersect at one point.

This is called the heart of a triangle.

Proximity theorem

The bisector of the inner corner of a triangle intersects the bisector of the outer corner at the other two vertices.

This is called the center of mass of a triangle. A triangle has three centers of mass.

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.

pappus law

The center line of the triangle is parallel to the third side and equal to half of the third side.

Trilateral relation theorem

The sum of any two sides of a triangle is greater than the third side, and the difference between any two sides is less than the third side.

pythagorean theorem

In Rt triangle ABC, A≤90 degrees, then

AB AB+AC AC=BC BC

A > 90 degrees, then

AB a b+ AC AC & gt； B.C.

Spartan king

Menelaus, an ancient Greek mathematician, first proved Menelaus theorem. It points out that if a straight line intersects with three sides AB, BC, CA of △ABC or its extension lines at points F, D and E, then (AF/FB )× (BD/DC )× (CE/EA) =1.

Prove:

The passing point a is the extension line where AG‖BC intersects DF at g,

Then af/FB = ag/BD, BD/DC = BD/DC, ce/ea = DC/ag.

Three formulas are multiplied: AF/FB× BD/DC× CE/EA = Ag/BD× BD/DC× DC/Ag =1.

The inverse theorem also holds: if there are three points F, D and E on the edges of AB, BC and CA or their extension lines, and they satisfy (AF/FB )× (BD/DC )× (CE/EA) =1,then these three points F, D and E are * * lines. Using this inverse theorem, we can judge the trisection line.

Cheval theorem

Let o be any point in △ABC,

AO, BO and CO intersect at D, E and F respectively, then BD/DC*CE/EA*AF/FB= 1.

Brief introduction of proof method

(i) This problem can be proved by Menelaus theorem:

Σ△ ADC is intercepted by a straight line BOE,

∴ CB/BD*DO/OA*AE/EC= 1 ①

△ABD is cut by straight COF, ∴ BC/CD*DO/OA*AF/BF= 1②.

② ①: BD/DC*CE/EA*AF/FB= 1。

(2) It can also be proved by the area relationship.

∫BD/DC = S△ABD/S△ACD = S△BOD/S△COD =(S△ABD-S△BOD)/(S△ACD-S△COD)= S△AOB/S△AOC③

Similarly, ce/ea = s △ BOC/s △ aob4af/FB = s △ AOC/s △ boc⑤.

③××× ⑤ BD/DC*CE/EA*AF/FB= 1。

Prove that three high lines of a triangle must intersect at one point by Seva theorem;

Let the vertical legs of three sides AB, BC and AC be D, E and F respectively.

According to the inverse theorem of Seva's theorem, because (AD: DB) * (BE: EC) * (CF: FA) = [(CD * CTGA)/[(CD * CTGB)] * [(AE * CTGB)/(AE * CTGC)] * [(BF * CTGC)/

[(AE*ctgB)]= 1, so the three heights of CD, AE and BF intersect at one point.