basic concept

Axiom 1: If two points on a straight line are in a plane, then all points on this straight line are in this plane.

Axiom 2: If two planes have a common point, then they have only one common straight line passing through this point.

Axiom 3: When three points that are not on a straight line intersect, there is one and only one plane.

Inference 1: Through a straight line and a point outside this straight line, there is one and only one plane.

Inference 2: Through two intersecting straight lines, there is one and only one plane.

Inference 3: Through two parallel straight lines, there is one and only one plane.

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

Equiangular Theorem: If two sides of one angle are parallel and in the same direction as two sides of another angle, then the two angles are equal.

The positional relationship between two straight lines in space;

There are only three positional relationships between two straight lines in space: parallel, intersecting and nonplanar.

1, according to whether * * * surface can be divided into two categories:

(1)*** plane: parallel intersection.

(2) Different planes:

Definition of non-planar straight lines: two different straight lines on any plane are neither parallel nor intersecting.

Judgment theorem of out-of-plane straight line: use the straight line between a point in the plane and a point out of the plane, and the straight line in the plane that does not pass through this point is the out-of-plane straight line.

The angle formed by two straight lines on different planes: the range is (0,90) esp. Space vector method

Distance between two straight lines in different planes: common vertical line segment (only one) esp. Space vector method

2, if from the perspective of the existence of public * * *, points can be divided into two categories:

(1) has only one thing in common-intersecting straight lines; (2) There is nothing in common-parallel or non-parallel.

The positional relationship between a straight line and a plane:

There are only three positional relationships between a straight line and a plane: within the plane, intersecting the plane and parallel to the plane.

(1) The straight line is in the plane-there are countless things in common.

(2) A straight line intersects a plane-there is only one common point.

Angle between a straight line and a plane: the acute angle formed by the diagonal of a plane and its projection on the plane.

Esp。 Space vector method (finding the normal vector of a plane)

Provisions: a, when the straight line is perpendicular to the plane, the angle formed is a right angle; B, when the line is parallel or in the plane, the angle is 0.

The included angle between the straight line and the plane is [0,90].

Minimum angle theorem: the angle formed by the diagonal line and the plane is the smallest angle between the diagonal line and any straight line in the plane.

Three Verticality Theorems and Inverse Theorems: If a straight line in a plane is perpendicular to the projection of a diagonal line in this plane, it is also perpendicular to this diagonal line.

Esp。 This line is perpendicular to the plane.

Definition of vertical line and plane: If straight line A is perpendicular to any straight line in the plane, we say that straight line A and plane are perpendicular to each other. The straight line A is called the perpendicular of the plane, and the plane is called the vertical plane of the straight line A. ..

Theorem for judging whether a straight line is perpendicular to a plane: If a straight line is perpendicular to two intersecting straight lines in a plane, then the straight line is perpendicular to the plane.

Theorem of the property that straight lines are perpendicular to a plane: If two straight lines are perpendicular to a plane, then the two straight lines are parallel.

③ The straight line is parallel to the plane-there is nothing in common.

Definition of parallelism between straight line and plane: If straight line and plane have nothing in common, then we say that straight line and plane are parallel.

Theorem for determining the parallelism between a straight line and a plane: If a straight line out of the plane is parallel to a straight line in this plane, then this straight line is parallel to this plane.

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.

The positional relationship between two planes:

(1) The definition that two planes are parallel to each other: there is no common point between two planes in space.

(2) the positional relationship between two planes:

The two planes are parallel-have nothing in common; Two planes intersect-there is a straight line.

First, parallel

Theorem for determining the parallelism of two planes: If two intersecting lines in one plane are parallel to the other plane, then the two planes are parallel.

Parallel theorem of two planes: if two parallel planes intersect with the third plane at the same time, the intersection lines are parallel.

B, crossroads

dihedral angle

(1) Half-plane: A straight line in a plane divides this plane into two parts, and each part is called a half-plane.

(2) dihedral angle: The figure composed of two half planes starting from a straight line is called dihedral angle. The range of dihedral angle is [0, 180].

(3) The edge of dihedral angle: This straight line is called the edge of dihedral angle.

(4) Dihedral facet: These two half planes are called dihedral facets.

(5) Plane angle of dihedral angle: Take any point on the edge of dihedral angle as the endpoint, and make two rays perpendicular to the edge in two planes respectively. The angle formed by these two rays is called the plane angle of dihedral angle.

(6) Straight dihedral angle: A dihedral angle whose plane angle is a right angle is called a straight dihedral angle.

Esp。 The two planes are perpendicular.

Definition of two planes perpendicular: two planes intersect, and if the angle formed is a straight dihedral angle, the two planes are said to be perpendicular to each other. Write it down as X.

A theorem to determine the perpendicularity of two planes: If one plane passes through the perpendicular of the other plane, then the two planes are perpendicular to each other.

Verticality theorem of two planes: If two planes are perpendicular to each other, a straight line perpendicular to the intersection in one plane is perpendicular to the other plane.

note:

Solution of dihedral angle: direct method (making plane angle), triple vertical theorem and inverse theorem, area projection theorem, normal vector method of space vector (pay attention to the complementary relationship between the obtained angle and the required angle)

polyhedron

prism

Definition of prism: two faces are parallel to each other, the other face is a quadrilateral, and the common sides of every two quadrilaterals are parallel to each other. The geometric shape enclosed by these faces is called a prism.

Properties of prism

(1) All sides are equal, and the sides are parallelogram.

(2) The sections parallel to the two bottom surfaces are congruent polygons.

(3) The cross section (diagonal plane) passing through two non-adjacent sides is a parallelogram.

pyramid

Definition of Pyramid: One face is a polygon and the other faces are triangles with a common vertex. The geometry surrounded by these faces is called a pyramid.

The essence of the pyramid:

The sides of (1) intersect at one point. The sides are triangular.

(2) The section parallel to the bottom surface is a polygon similar to the bottom surface. And its area ratio is equal to the square of the ratio of the height of the truncated pyramid to the height of the far pyramid.

Positive pyramid

Definition of a regular pyramid: If the bottom of the pyramid is a regular polygon and the projection of the vertex at the bottom is the center of the bottom, such a pyramid is called a regular pyramid.

The nature of the regular pyramid:

(1) An isosceles triangle whose sides intersect at one point and are equal. The height on the base of each isosceles triangle is equal, which is called the oblique height of a regular pyramid.

(3) Some special right-angled triangles

esp:

A For a regular triangular pyramid with two adjacent sides perpendicular to each other, the projection of the vertex on the bottom surface can be obtained as the vertical center of the triangle on the bottom surface by the three perpendicular theorems.

B there are three pairs of straight lines with different planes in the tetrahedron. If two pairs are perpendicular to each other, the third pair is perpendicular. And the projection of the vertex on the bottom surface is the vertical center of the triangle on the bottom surface.

Linear sum equation

(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.

At that time,;

At that time,;

It didn't exist then.

② Slope formula of straight line passing through two points:

Pay attention to the following four points:

(1) At that time, the right side of the formula was meaningless, the slope of the straight line did not exist, and the inclination angle was 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. However, because

The abscissa of each point on L is equal to x 1, so 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) Cutting torque type:

Where the straight line intersects the axis at the point and intersects the axis at the point, that is, the intercepts with the axis and the axis are respectively.

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

Note: Various scope of application.

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) Vertical linear system

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

(3) A linear system passing through a fixed point

① Straight line system with slope k: a straight line passes through a fixed point;

② The equation of the line system at the intersection of two straight lines is

(is 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

stride

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 a plane rectangular coordinate system,

rule

(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 the point to the straight line to solve it.

Equation of circle

(1) standard equation, center and radius r;

(2) General equation

At that time, the equation represented a circle. At this point, the center is and the radius is.

At that time, I said 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. Determining a circle requires three independent conditions. If the standard equation of a circle is used,

Demand a, b, 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.

The positional relationship between straight line and circle

The positional relationship between a straight line and a circle includes three situations: separation, tangency and intersection:

(1) Set a straight line and a circle, and the distance from the center of the circle to L is, then there is; ;

(2) Tangent to a point outside the circle: ①k does not exist, so verify the existence of ②k, establish an oblique equation, and solve k with the distance from the center of the circle to the straight line = radius, and get two solutions of the equation.

(3) The tangent equation of a point passing through a circle: circle (x-a)2+(y-b)2=r2, and a point on the circle is (x0, y0), then the tangent equation passing through that point is (x0-a) (x-a)+(y0-b) (y-b) =

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.

At that time, the two circles were separated, and there were four common tangents at this time;

At that time, the two circles were circumscribed, and the connection line crossed the tangent point, with two outer tangents and one inner common tangent;

At that time, the two circles intersect, and the connecting line bisects the common chord vertically, and there are two external tangents;

At that time, two circles were inscribed, and the connecting line passed through the tangent point, and there was only one common tangent;

At that time, two circles included; It was concentric circles.

Note: when two points on the circle are known, the center of the circle must be on the vertical line in the middle; It is known that two circles are tangent and two centers are tangent to the tangent point.

The auxiliary line of a circle generally connects the center of the circle with the tangent or the midpoint of the chord of the center of the circle.