I. Lines and equations

(1) inclination angle of straight line

Definition: The angle between the positive direction of the X axis and the upward direction of the straight line is called the inclination angle of the straight line. In particular, when a straight line is parallel or coincident with the X axis, we specify that its inclination angle is 0 degrees. Therefore, the range of inclination angle is 0 ≤α < 180.

(2) the slope of the straight line

① Definition: A straight line whose inclination is not 90, and the tangent of its inclination is called the slope of this straight line. The slope of a straight line is usually represented by k, that is. Slope reflects the inclination of straight line and axis. When. When,; When, does not exist.

② Slope formula of straight line passing through two points:

Pay attention to the following four points: (1) When the right side of the formula is meaningless, the slope of the straight line does not exist, and the inclination angle is 90;

(2)k has nothing to do with the order of P 1 and P2;

(3) The slope can be obtained directly from the coordinates of two points on a straight line without inclination angle;

(4) To find the inclination angle of a straight line, we can find the slope from the coordinates of two points on the straight line.

(3) Linear equation

① Point-oblique type: the slope of the straight line is k, passing through the point.

Note: When the slope of the straight line is 0, k=0, and the equation of the straight line is y=y 1. When the slope of the straight line is 90, the slope of the straight line does not exist, and its equation can not be expressed by point inclination. But because the abscissa of each point on L is equal to x 1, its equation is x=x 1.

② Oblique section: the slope of the straight line is k, and the intercept of the straight line on the Y axis is b..

③ Two-point formula: () Two points on a straight line,

④ Intercept formula: where the straight line intersects with the axis at the point and intersects with the axis at the point, that is, the intercepts with the axis and the axis are respectively.

⑤ General formula: (A and B are not all 0)

Note: ○ 1 scope of application.

○2 Special equations such as: straight line parallel to X axis: (b is constant); A straight line parallel to the Y axis: (A is a constant);

(4) Linear system equation: that is, a straight line with some * * * property.

(1) parallel linear system

A linear system parallel to a known straight line (a constant that is not all zero): (c is a constant)

(2) A linear system passing through a fixed point

(i) Linear system with slope k: a straight line passes through a fixed point;

(2) The equation of the straight line system where two straight lines intersect is (as a parameter), where the straight line is not in the straight line system.

(5) Two straight lines are parallel and vertical.

When, when,;

Note: When judging the parallelism and verticality of a straight line by using the slope, we should pay attention to the existence of the slope.

(6) The intersection of two straight lines

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The coordinates of the intersection point are a set of solutions of the equation. These equations have no solution; The equation has many solutions and coincidences.

(7) Distance formula between two points: Let it be two points in the plane rectangular coordinate system, then

(8) Distance formula from point to straight line: distance from point to straight line.

(9) Distance formula of two parallel straight lines: take any point on any straight line, and then convert it into the distance from that point to the straight line.

Second, the equation of circle

1. Definition of a circle: The set of points whose distance to a point on a plane is equal to a fixed length is called a circle, the fixed point is the center of the circle, and the fixed length is the radius of the circle.

2. Equation of circle

(1) standard equation, center and radius r;

(2) General equation

When the equation represents a circle. At this point, the center is and the radius is.

When represents a point; At that time, the equation did not represent any graph.

(3) Method of solving cyclic equation:

Generally, the undetermined coefficient method is adopted: first set, then seek. Determine a circle requires three independent condition,

If you use the standard equation of a circle, you need a, b and r; If you use general equations, you need to find d, e, f;

In addition, we should pay more attention to the geometric properties of the circle: for example, the vertical line of a chord must pass through the origin, so as to determine the position of the center of the circle.

3, the position relationship between straight line and circle:

There are three situations in the positional relationship between a straight line and a circle: separation, tangency and intersection, which are basically judged by the following two methods:

(1) Set a straight line, and the distance from the center of the circle to L is 0.

(2) Set a straight line and a circle, first set up the elimination equation, and get a quadratic equation with one variable, so that the discriminant is, then there is; ;

Note: If the center of the circle is at the origin, we can use the formula to solve the tangent problem between a straight line and a circle, where the tangent point coordinates and r represent the radius.

(3) The tangent equation of a point on the circle:

① The circle x2+y2=r2, and a point on the circle is (x0, y0), then the tangent equation passing through this point is (textbook proposition).

② If the circle (x-a)2+(y-b)2=r2, and a point on the circle is (x0, y0), then the tangent equation passing through the point is (x0-a)(x-a)+(y0-b)(y-b)= r2 (the generalization of the textbook proposition).

4. The positional relationship between circles: it is determined by comparing the sum (difference) of the radii of two circles with the distance (d) between the center of the circle.

Set a circle,

The positional relationship between two circles is usually determined by comparing the sum (difference) of the radii of the two circles with the distance (d) between the center of the circle.

When two circles are separated, there are four common tangents at this time;

When two circles are tangent, the connecting line passes through the tangent point, and there are two external tangents and one internal common tangent;

When two circles intersect, the connecting line bisects the common chord vertically and has two external tangents;

When two circles are inscribed, the connecting line passes through the tangent point and there is only one common tangent;

When, two circles contain; When, for concentric circles.

Third, preliminary solid geometry

Structural characteristics of 1, column, cone, platform and ball

(1) prism:

Definition: Geometry surrounded by two parallel faces, the other faces are quadrangles, and the common edges of every two adjacent quadrangles are parallel to each other.

Classification: According to the number of sides of the bottom polygon, it can be divided into three prisms, four prisms and five prisms.

Representation: Use the letter of each vertex, such as a five-pointed star, or use the letter at the opposite end, such as a five-pointed star.

Geometric features: the two bottom surfaces are congruent polygons with parallel corresponding sides; The lateral surface and diagonal surface are parallelograms; The sides are parallel and equal; The section parallel to the bottom surface is a polygon that is congruent with the bottom surface.

② Pyramid

Definition: One face is a polygon, and the other faces are triangles with a common vertex. These faces enclose a geometric figure.

Classification: According to the number of sides of the bottom polygon, it can be divided into three pyramids, four pyramids and five pyramids.

Representation: Use the letters of each vertex, such as a pentagonal pyramid.

Geometric features: the side and diagonal faces are triangles; The section parallel to the bottom surface is similar to the bottom surface, and its similarity ratio is equal to the square of the ratio of the distance from the vertex to the section to the height.

(3) Prism:

Definition: Cut off the part between the pyramid, the section and the bottom with a plane parallel to the bottom of the pyramid.

Classification: According to the number of sides of the bottom polygon, it can be divided into triangular, quadrangular and pentagonal shapes.

Representation: Use the letters of each vertex, such as a pentagonal pyramid.

Geometric features: ① The upper and lower bottom surfaces are similar parallel polygons; ② The side is trapezoidal; ③ The sides intersect with the vertices of the original pyramid.

(4) Cylinder:

Definition: Geometry surrounded by a surface with one side of a rectangle and the other three sides rotating around a straight line.

Geometric features: ① The bottom is an congruent circle; ② The bus is parallel to the shaft; ③ The axis is perpendicular to the radius of the bottom circle; ④ The side development diagram is a rectangle.

(5) Cone:

Definition: Rotate the geometry surrounded by the surface of Zhou Suocheng with the right-angled side of the right-angled triangle as the rotation axis.

Geometric features: ① the bottom is round; (2) The generatrix intersects with the apex of the cone; ③ The side spread diagram is a fan.

(6) frustum of a cone:

Definition: Cut the part between the cone, the section and the bottom with a plane parallel to the bottom of the cone.

Geometric features: ① The upper and lower bottom surfaces are two circles; (2) The side generatrix intersects with the vertex of the original cone; (3) The side development diagram is an arch.

(7) Sphere:

Definition: Geometry formed by taking the straight line where the diameter of the semicircle is located as the rotation axis and the semicircle surface rotates once.

Geometric features: ① the cross section of the ball is round; ② The distance from any point on the sphere to the center of the sphere is equal to the radius.

2. Three views of space geometry

Define three views: front view (light is projected from the front of the geometry to the back); Side view (from left to right) and top view (from top to bottom)

Note: the front view reflects the position relationship of the object, that is, it reflects the height and length of the object;

The top view reflects the position relationship between the left and right of the object, that is, the length and width of the object;

The side view reflects the up-and-down and front-and-back positional relationship of the object, that is, it reflects the height and width of the object.

3. Intuition of space geometry-oblique two-dimensional drawing method.

The characteristics of oblique bisection method are as follows: ① The line segment originally parallel to the X axis is still parallel to X, and its length remains unchanged;

② The line segment originally parallel to the Y axis is still parallel to Y, and its length is half of the original.

4. Surface area and volume of cylinders, cones and platforms.

The surface area of a (1) geometry is the sum of all the surfaces of the geometry.

(2) The surface area formula of special geometry (C is the perimeter of the bottom, H is the height, and L is the generatrix)

(3) Volume formulas of cylinders, cones and platforms.

(4) Formula of surface area and volume of sphere: v =；; S =

5, spatial point, straight line, plane position relationship

(1) plane

① The concept of plane: a. Descriptive description; B. infinite extension of plane;

2 plane representation: usually expressed by Greek letters α, β and γ, such as plane α (usually written as acute angle); It can also be represented by two letters with opposite vertices, such as plane BC.

③ Relationship between point and plane: point A is in the plane, and it is recorded as; The point is not on the plane, recorded as

The relationship between point and straight line: on the straight line L of point A, it is recorded as: a ∈ l; Point a is outside the straight line l and marked as A l；; ;

The relationship between the straight line and the plane: the straight line L is in the plane α, which is denoted as L α; The straight line l is not in the plane α, and it is recorded as l α.

(2) Axiom 1: If two points of a straight line are in a plane, then all points of the straight line are in this plane. (that is, the straight line is in the plane, or the plane passes through the straight line)

Application: check whether the desktop is flat; Judge whether the straight line is in the plane. Express axiom1in symbolic language;

(3) Axiom 2: After passing through three points that are not on the same straight line, there is only one plane.

Inference: a straight line and a point outside the straight line determine a plane; Two intersecting straight lines define a plane; Two parallel straight lines define a plane.

Axiom 2 and its reasoning function: ① it is the basis for determining planes in space ② it is the basis for proving plane coincidence.

(4) Axiom 3: If two non-coincident planes have a common point, then they only have a common straight line passing through the point.

Symbol: Plane α and β intersect, the intersection line is A, and it is denoted as α ∩ β = A. Symbol language:

The function of axiom 3: ① is the method to judge the intersection of two planes.

② Explain the relationship between the intersection of two planes and the common point of two planes: the intersection must pass through the common point.

(3) It can be judged that a point is on a straight line, which is an important basis for proving several points.

(5) Axiom 4: Two lines parallel to the same line are parallel to each other.

(6) The positional relationship between spatial straight lines.

① Definition of non-planar straight lines: two straight lines that are different from each other on any plane.

② Properties of straight lines in different planes: neither parallel nor intersecting.

③ Determination of out-of-plane straight line: A straight line passing through a point out of plane and a point in plane is an out-of-plane straight line.

(4) Angle formed by straight lines of different planes: Lines A and B are straight lines of different planes. When passing through any point O in space, straight lines A '∨A and B '∨B are introduced respectively, and the acute angle (or right angle) formed by straight lines A' and B' is called the angle formed by straight lines A and B. The angle range formed by two different planes is (0,90). If the angle formed by straight lines of two different planes is a right angle, we say that the straight lines of two different planes are perpendicular to each other.

Description: (1) Methods for judging whether a spatial straight line is an out-of-plane straight line: ① According to the definition of an out-of-plane straight line; (2) The judging theorem of straight lines in different planes.

(2) In the definition of the angle formed by straight lines in different planes, the point O in space is arbitrary, regardless of the position of the point O. ..

(3) the step of finding the angles formed by straight lines on different planes:

A, using the defined structural angle, one can be fixed, the other can be translated, or both can be translated to a special position at the same time, and the vertex can be selected at a special position.

B, prove that the angle is the angle.

C, use triangle to find the angle

(7) Equiangular theorem: If two sides of one angle and two sides of another angle are parallel respectively, then the two angles are equal or complementary.

(8) The positional relationship between spatial straight line and plane.

A straight line is in a plane-there are countless things in common.

Symbolic representation of three positional relationships: aα a ∩ α = aα a ∩ α.

(9) positional relationship between planes: parallel-no common point; α ∑ β Intersection-There is a common straight line. α∪β= b

6. Parallel problems in space

Determination and properties of (1) parallel lines and planes

Theorem for judging the parallelism between a straight line and a plane: A straight line out of the plane is parallel to a straight line in the plane, then the straight line is parallel to the plane. Parallel lines, parallel lines, parallel planes.

Theorem of parallelism between straight lines and planes: If a straight line is parallel to a plane and the plane passing through it intersects with this plane, then the straight line is parallel to the intersection line.

Parallel lines, parallel lines, parallel lines.

(2) The judgment and nature of parallelism between planes.

Theorem for judging the parallelism of two planes (1) If two intersecting straight lines in one plane are parallel to the other plane, then the two planes are parallel (line-plane parallel → plane-plane parallel).

(2) If two sets of intersecting straight lines are parallel in two planes, the two planes are parallel. (parallel lines → parallel planes),

(3) Two planes perpendicular to the same straight line are parallel,

Parallelism Theorem of Two Planes (1) If two planes are parallel, then a straight line in one plane is parallel to the other plane. (Face-to-face parallelism → Line-to-Line parallelism)

(2) If two parallel planes intersect with the third plane, their intersection lines are parallel. (Face-to-face parallelism → Line-to-Line parallelism)

7. Vertical problem in space

(1) Definition of line, surface and line-surface verticality

(1) Perpendicularity of two straight lines with different planes: If the angle formed by two straight lines with different planes is a right angle, the two straight lines with different planes are said to be perpendicular to each other.

② Line-plane verticality: If a straight line is perpendicular to any straight line in a plane, it is said that the straight line is perpendicular to the plane.

③ Plane is perpendicular to the plane: if two planes intersect, the dihedral angle (the figure formed by two half planes starting from a straight line) is a straight dihedral angle (the plane angle is a right angle), which means that the two planes are perpendicular.

(2) Determination of vertical relation and property theorem.

(1) The judging theorem and property theorem of the perpendicularity between a straight line and a plane.

Decision theorem: If a straight line is perpendicular to two intersecting straight lines on a plane, then the straight line is perpendicular to the plane.

Property theorem: If two straight lines are perpendicular to a plane, then the two straight lines are parallel.

(2) The judgment theorem and property theorem of vertical plane.

Decision theorem: If one plane passes through the vertical line of the other plane, then the two planes are perpendicular to each other.

Theorem of nature: If two planes are perpendicular to each other, then the straight line perpendicular to their intersection on one plane is perpendicular to the other plane.

8. Spatial perspective

(1) Angle between straight lines

① Angle formed by two parallel straight lines: specified as.

(2) The angle formed by the intersection of two straight lines: the angle formed by the intersection of two straight lines is not greater than the right angle, which is called the angle formed by these two straight lines.

(3) Angle formed by two straight lines with different planes: when passing through any point o in space, make the straight line parallel to the two straight lines with different planes A and B to form two intersecting straight lines, and the angle formed by these two intersecting straight lines is called the angle formed by two straight lines with different planes.

(2) The angle formed by a straight line and a plane

① The angle formed by the parallel lines between the plane and the plane: specified as.

② The angle between the plane and the perpendicular to the plane: specified as.

(3) The angle formed by the oblique line of the plane and the plane: the acute angle formed by an oblique line of the plane and its projection in the plane is called the angle formed by this straight line and this plane.

The idea of finding the angle between diagonal and plane is similar to finding the angle formed by straight lines on different planes: "one work, two certificates and three calculations"

When making an angle, project according to the definition key. From the definition of projection, the key lies in the point on the diagonal to the perpendicular to the surface.

When solving a problem, pay attention to mining two pieces of information in the problem setting: (1) a point on the diagonal is perpendicular to the surface; (2) The diagonal or a point on the plane of the diagonal is perpendicular to the known surface, and the vertical line can be easily obtained from the vertical nature of the surface.

(3) The dihedral angle of dihedral angle and plane angle

① Definition of dihedral angle: The figure formed by two half planes starting from a straight line is called dihedral angle, this straight line is called the edge of dihedral angle, and these two half planes are called the faces of dihedral angle.

② Plane angle of dihedral angle: Take any point on the edge of dihedral angle as the vertex and make two rays perpendicular to the edge in two planes. The angle formed by these two rays is called the plane angle of dihedral angle.

③ Straight dihedral angle: A dihedral angle whose plane angle is a right angle is called a straight dihedral angle. If the dihedral angle formed by two intersecting planes is a straight dihedral angle, then the two planes are vertical; On the contrary, if two planes are perpendicular, the dihedral angle formed is a straight dihedral angle.

(4) Calculation method of dihedral angle

Definition method: select the relevant point on the edge, and make a ray perpendicular to the edge in two planes through this point to get the plane angle.

Vertical plane method: when the vertical lines from one point to two surfaces in dihedral angle are known, the angle formed by the intersection of two vertical lines as the intersection of plane and two surfaces is the plane angle of dihedral angle.

9. Spatial Cartesian Coordinate System

(1) Definition: As shown in the figure, it is a unit cube with A as the origin and OD, O and OB as the positive directions respectively.

Create three number axes. At this time, the spatial rectangular coordinate system Oxyz is established.

1)O is called coordinate origin 2)x axis, Y axis and Z axis are called coordinate axes 3) The plane passing through every two coordinate axes is called coordinate plane.

(2) Representation of the right hand: the possible position of the thumb, forefinger and middle finger of the right hand when they are perpendicular to each other. The thumb points to the positive direction of the X axis, the index finger points to the positive direction of the Y axis, and the middle finger points to the positive direction of the Z axis, and the phase between the three axes can also be determined.

(3) Coordinate representation of any point: the coordinate of a point M in space can be represented by an ordered real array, which is called the coordinate of the point M in the rectangular coordinate system of the space, and is denoted as (X is called the abscissa of the point M, Y is called the ordinate of the point M, and Z is called the ordinate of the point M).

(4) The coordinate formula of the distance between two points in space:

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