What are the rules of triangles?
Properties of Triangles
Angle
1 The sum of the interior angles of a triangle on the plane is equal to 180° (Sum of Interior Angle Theorem);
2 In The sum of the exterior angles of a triangle on the plane is equal to 360° (Exterior Angle Sum Theorem);
3 The exterior angle of a triangle on the plane is equal to the sum of its two non-adjacent interior angles.
Corollary: An exterior angle of a triangle is greater than any interior angle that is not adjacent to it.
4. At least two of the three interior angles of a triangle are acute.
5 In the triangle, at least one angle is greater than or equal to 60 degrees, and at least one angle is less than or equal to 60 degrees.
Sides
6 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.
7 In a right triangle, if one angle is equal to 30 degrees, then the right side opposite to the 30-degree angle is half the hypotenuse.
8 The sum of the squares of the two right-angled sides of a right triangle is equal to the square of the hypotenuse (Pythagorean theorem).
*Converse theorem of the Pythagorean theorem: If the lengths of the three sides of a triangle a, b, and c satisfy a?+b?=c?, then the triangle is a right-angled triangle.
9 The midline of the hypotenuse of a right triangle is equal to half of the hypotenuse.
10 The three angle bisectors of a triangle intersect at one point, the straight lines of the three altitude lines intersect at one point, and the three midlines intersect at one point.
11 The sum of the squares of the lengths of the three midlines of a triangle is equal to 3/4 of the sum of the squares of the lengths of its three sides.
12 Triangles with equal bases and equal heights have equal areas.
13 The ratio of the areas of triangles with equal bases is equal to the ratio of their heights, and the ratio of the areas of triangles with equal heights is equal to the ratio of their bases.
14 Any midline of a triangle divides the triangle into two triangles with equal areas.
15 The angle bisector of the vertex of an isosceles triangle, the height of the base and the midline of the base are on a straight line (three lines combined into one). What are the rules of Yang Hui's triangle?
1. Each row of numbers is symmetrical, starting from 1 and gradually getting larger, then getting smaller, and returning to 1. 2. The number of numbers in the nth row is n. 3. The sum of the numbers in the nth row is 2^(n-1). 4. Each number is equal to the sum of the left and right numbers in the previous row. The entire Pascal triangle can be written using this property.
5. Connect the first number in row 2n+1, the third number in row 2n+2, the fifth number in row 2n+3... into a line. The sum of these numbers is the 2nth Fibonacci number. Take the 2nd number in row 2n, the 4th number in row 2n+1, the 6th number in row 2n+2...the sum of these numbers is the 2n-1 Fibonacci number. 6. The first number in the nth row is 1, the second number is 1×(n-1), the third number is 1×(n-1)×(n-2)/2, and the fourth number is 1×(n-1)×(n-2)/2. The number is 1×(n-1)×(n-2)/2×(n-3)/3…and so on.
Yang Hui's triangle is a triangular number table arranged by numbers. The general form is as follows:
1 n=0
1 1 n=1
1 2 1 n=2
1 3 3 1 n=3
1 4 6 4 1 n=4
1 5 10 10 5 1 n=5
1 6 15 20 15 6 1 n=6
…
The numbers in each row of this sequence are exactly binomials After a + b is raised to the power, expand the coefficients of each term.
For example:
(a+b)^1=a^1+b^1
(a+b)^2=a^2+2ab+b^2
(a+b)^3=a^3+3a^2b+3ab^2+b^3
……
(a+b)^ 6=a^6+6a^5b+15a^4b^2+20a^3b^3+15a^2b^4+6ab^5+b^6 (pay attention to discover the rules)
…… < /p>
What are the rules for counting line segments, angles, and triangles?
To count line segments, count the number of points on the line segment, including the end points, and divide the difference between the number of points by the number of points minus one by two, that is: n×﹙n-1﹚÷2 Number The formulas for counting angles and line segments are the same. What are the rules for solving triangles
Do the questions yourself and pay attention to the summary;
The midline on one side of the triangle is cut into 2:1 by the other midline. Proportion, the length of the side close to the vertex
The triangle is divided into two triangles with equal areas
What are the rules for the sum of the interior angles of a parallelogram and other triangles by extending the midline?
The rule is: the sum of the interior angles of a triangle is 180°. Are there any rules for the triangle rule?
The center of gravity, circumcenter, perpendicular center, incenter and paracenter of a triangle are called the five centers of the triangle. The five-center theorem of a triangle refers to the general name of the triangle's centroid theorem, circumcenter theorem, perpendicular theorem, inner center theorem and paracenter theorem.
2. Center of Gravity Theorem
The midlines of the three sides of a triangle intersect at one point. This point is called the center of gravity of the triangle. It can be proved by the Swallowtail Theorem that the three midlines intersect at one point, which is very simple. (The center of gravity is originally a physical concept. For a triangular sheet of equal thickness and uniform mass, the center of gravity is exactly the intersection of the three center lines of the triangle, hence the center of gravity.)
Properties of the center of gravity:
< p> 1. The ratio of the distance from the center of gravity to the vertex and the distance from the center of gravity to the midpoint of the opposite side is 2:1.2. The areas of the three triangles formed by the center of gravity and any two vertices of the triangle are equal. That is, the distance from the center of gravity to the three sides is inversely proportional to the length of the three sides.
3. The sum of the squares of the distances from the center of gravity to the three vertices of the triangle is the smallest.
4. In the plane rectangular coordinate system, the coordinates of the center of gravity are the arithmetic mean of the coordinates of the vertices, that is, the coordinates of the center of gravity are ((X1+X2+X3)/3, (Y1+Y2 +Y3)/3).
5. The sum of the three vectors starting from the center of gravity and ending with the three vertices of the triangle is equal to zero vector.
3. The circumcenter theorem
The center of the circumcircle of a triangle is called the circumcenter of the triangle.
Properties of the circumcenter:
1. The perpendicular bisectors of the three sides of a triangle intersect at a point, which is the circumcenter of the triangle.
2. If O is the circumcenter of △ABC, then ∠BOC=2∠A (∠A is an acute or right angle) or ∠BOC=360°-2∠A (∠A is an obtuse angle).
3. When the triangle is an acute triangle, the circumcenter is inside the triangle; when the triangle is an obtuse triangle, the circumcenter is outside the triangle; when the triangle is a right triangle, the circumcenter is on the hypotenuse, and The midpoints of the hypotenuses coincide.
4. To calculate the coordinates of the circumcenter, you should first calculate the following temporary variables: d1, d2, and d3 are the point products of the vectors connecting the three vertices of the triangle to the other two vertices. c1=d2d3, c2=d1d3, c3=d1d2; c=c1+c2+c3. Circentric coordinates: ((c2+c3)/2c, (c1+c3)/2c, (c1+c2)/2c).
5. The distances from the circumcenter to the three vertices are equal
4. Vertical centroid theorem
The three heights (straight lines) of a triangle intersect at one point, which is called a triangle of heart.
Properties of the vertical center:
1. A triangle has three vertices, three vertical feet, and these 7 points of the vertical center can produce 6 four-point circles.
2. The circumcenter O, center of gravity G and vertical center H of the triangle are three-point *** lines, and OG: GH = 1: 2. (This straight line is called the Euler line of the triangle)
3. The distance from the vertical center to a vertex of the triangle is twice the distance from the circumcenter of the triangle to the side opposite the vertex.
4. The product of the two parts of each vertical line divided vertically is equal.
Proof of theorem
Known: In ΔABC, AD and BE are two heights, AD and BE intersect at point O, connect line CO and extend to intersect AB at point F. Verify ：CF⊥AB
Proof:
Connect the line DE ∵∠ADB=∠AEB=90 degrees ∴A, B, D, E four-point *** circle ∴∠ADE= ∠ABE
∵∠EAO=∠DAC ∠AEO=∠ADC ∴ΔAEO∽ΔADC
∴AE/AO=AD/AC ∴ΔEAD∽ΔOAC ∴∠ACF=∠ADE= ∠ABE
And ∵∠ABE+∠BAC=90 degrees ∴∠ACF+∠BAC=90 degrees ∴CF⊥AB
Therefore, the vertical center theorem holds!
5 Incenter Theorem
The center of the inscribed circle of a triangle is called the incenter of the triangle.
Nature of the heart:
1. The three interior angle bisectors of a triangle intersect at one point. This point is the center of the triangle.
2. The distance from the incenter to the sides of a right triangle is equal to one-half the difference between the sum of the two right-angled sides minus the hypotenuse.
3. P is any point in the space where ΔABC is located, and the necessary and sufficient condition that point 0 is the center of ΔABC is: vector P0=(a×vector PA+b×vector PB+c×vector PC)/( a+b+c).
4. O is the center of the triangle, A, B, and C are the three vertices of the triangle respectively. If AO is extended to intersect side BC at N, then AO:ON=AB :BN=AC:CN=(AB+AC):BC
5. (Eula’s Theorem) ⊿ABC, R and r are the radii of the circumscribed circle and the inscribed circle respectively, O and I Their outer center and inner center respectively, then OI^2=R^2-2Rr.
6. (The relationship between the lengths of the interior angle bisectors and the three sides)
In △ABC, 0 is the inner center, and the interior angle bisectors of ∠A, ∠B, and ∠C intersect BC and AC, AB are in Q, P, R, then BQ/QC=c/b, CP/PA=a/c, BR/RA=a/b.
7. Distance from the center to the three sides of the triangle equal.
6 Paracenter Theorem
The center of the circumcenter of a triangle (the circle tangent to one side of the triangle and the extension of the other two sides) is called the circumcenter of the triangle.
Properties of the circumcenter:
1. If the bisector of one interior angle of a triangle intersects with the bisectors of the exterior angles at the other two vertices, that point is the circumcenter of the triangle.
2. Each triangle has three circumcenters.
3. The distances from the pericenter to the three sides are equal.
As shown in the figure, point M is the center of △ABC. The intersection of the exterior angle bisectors of any two angles of a triangle and the interior angle bisector of the third angle. A triangle has three circumcenters, and they must be outside the triangle.
Attachment: The center of a triangle: Only an equilateral triangle has a center. At this time, the center of gravity, the inner center, the outer center, and the vertical center are all in one.
7 Related Poems
The Song of the Five Hearts of a Triangle (the emphasis is on the outside and the inside)
The triangle has five hearts, the emphasis is on the outside and the inside and the sides. Five The nature of the mind is very important. Master it carefully and don’t forget it.
Center of Gravity
The three center lines must intersect. The location of the intersection is really clever. The intersection point is named "center of gravity". The nature of the center of gravity must be clear.
The center of gravity divides the center line segments, and the number The ratio of paragraphs can be understood by ear; the ratio of length to short is two to one, and it can be mastered flexibly.
Circumcenter
A triangle has six elements, and its three interior angles have three sides. Draw the center perpendicular of the three sides, and the three lines intersect at one point.
This point is defined as the circumcenter, and it can be used to draw a circumscribed circle. Don’t forget to mix your inner heart and outer heart. Cutting inside and connecting outside is the key.
Vertical center
Draw three heights on the triangle. The three heights must intersect at the vertical center. The high line divides the triangle, and three right-angled pairs appear.
There are twelve right-angled triangles, forming six pairs of similar shapes. There are four-point *** circle diagrams, and you can find them clearly through careful analysis.
Inner heart
A triangle corresponds to three vertices, and each corner has a bisector. The intersection of the three lines determines a point, which is called "inner heart" and has its origin;
From the point to the three sides, it is equal The distance can be used as a circle inscribed in a triangle. The center of this circle is called the "center". This definition is natural.
It’s really good to remember the nature of the five hearts. Are there any rules for preschool children to count triangles?
My son can count one by one. What are the rules for drawing several right-angled triangles in a circle?
Answer: The rules for drawing several right-angled triangles in a circle:
Take any point C on the circumference of the circle and move it toward any part of the circle. Draw two lines AC and BC connecting the two ends of a diameter AB, and the drawn △ABC is a right triangle. The hypotenuse is the diameter AB, and AC and BC are two right-angled sides.
In primary school mathematics, what are the rules for counting triangles?
Take "A large triangle is composed of 100 identical small equilateral triangles. How many triangles are there in this large triangle?" as an example.
100 identical equilateral triangles are arranged into a 10-story large triangle.
(1) Count the number of single triangles 10×10=100.
(2) Count the number of triangles with pointed upwards. Calculating the number of triangles with pointed upwards also uses the method of adding integers backwards. The key is to determine the leading number. First count the number of upward-pointing triangles composed of 4 small triangles: start from 9 (9 is the first number, the first number is 1 less than the number of layers) and add backwards until you add to 1 (9+8+7+ 6+5+4+3+2+1=45); then count the number of upward-pointing triangles composed of 9 small triangles: start from 8 and add backwards until you add to 1 (8+7+6 +5+4+3+2+1=36); then count the number of upward-pointing triangles composed of 16 small triangles: start from 7 and add backwards until you reach 1, (7+6+5+ 4+3+2+1=28); then count the number of upward-pointing triangles composed of 25 small triangles: start from 6 and add backwards until you add to 1 (6+5+4+3+2 +1=21)... Calculate the number of upward-pointing triangles composed of 81 small triangles: start from 2 and add backwards until you add to 1 (2+1=3); finally calculate from 100 The number of upward-pointing triangles composed of 1 small triangle. (3) Count the number of inverted triangles. Starting from the inverted triangle composed of 4 small triangles, we also use the backward addition method. The key is to know what the leading number is. We will tell you the ratio of the leading number to the total of the inverted triangle composed of 4 small triangles. The number is 3 less, that is, 10-3=7, and the number is: 7+6+5+4+3+2+1=28. From now on, when calculating the number of inverted triangles, the leading number will be less than 2 each time. That is to say, the number of inverted triangles composed of 9 small triangles should start from 5 and add up backwards, that is, 5+4+3+2+1=15. , the number of inverted triangles composed of 16 small triangles should start from 3 and add up backwards, that is, 3+2+1=6. The number of inverted triangles composed of 25 small triangles is 1, and it is impossible to have An inverted triangle composed of 36, 49, or even more small triangles. (4) Add the number of single triangles, regular triangles and inverted triangles together to get the total number of triangles.