1。 The complementary angles of the same angle (or equal angle) are equal.

3。 The vertex angles are equal.

5。 The outer angle of a triangle is equal to the sum of two non-adjacent inner angles.

6。 Two straight lines perpendicular to the same straight line in the same plane are parallel lines.

7。 The same angle is equal and two straight lines are parallel.

12。 The bisector of the top angle, the height on the bottom edge and the midline on the bottom edge of the isosceles triangle coincide with each other.

16。 In a right triangle, the center line of the hypotenuse is equal to half of the hypotenuse.

19。 A point on the bisector of an angle is equal to the distance on both sides of the angle. And its inverse theorem.

2 1。 The parallel segments sandwiched between two parallel lines are equal. The vertical segments sandwiched between two parallel lines are equal.

22。 A set of parallelograms with parallel and equal opposite sides, or two sets of opposite sides are equal respectively, or the diagonal is bisected.

24。 A quadrilateral with three right angles and a parallelogram with the same diagonal are rectangles.

25。 Diamond nature: four sides are equal, diagonal lines are perpendicular to each other, and each diagonal line bisects a set of diagonal lines.

27。 The four corners of a square are right angles and the four sides are equal. The two diagonals are equal and bisected vertically, and each diagonal bisects a set of diagonals.

34。 If a pair of two central angles, two arcs, two chords, and the center distance between two chords are equal in the same circle or in the same circle, other corresponding quantities are equal.

36。 The diameter perpendicular to the chord bisects the chord and bisects the arc opposite to the chord. The diameter (not the diameter) that bisects the chord is perpendicular to the chord and bisects the arc opposite the chord.

43。 The two right triangles divided by the high line on the hypotenuse are similar to the original triangle.

46。 The ratio of similar triangles to the high line, the ratio to the center line and the ratio to the angular bisector are all equal to the similarity ratio. The ratio of similar triangles area is equal to the square of similarity ratio.

37. The diagonals of the quadrilateral inscribed in the circle are complementary, and any external angle is equal to its internal angle.

47。 The judgment theorem of tangent passes through the outer end of the radius, and the straight line perpendicular to this radius is the tangent of the circle.

48。 The property theorem of tangent ① A straight line whose center is perpendicular to the tangent must pass through the tangent point. ② The tangent of the circle is perpendicular to the radius passing through the tangent point. ③ The straight line perpendicular to the tangent point must pass through the center of the circle.

49。 The tangent length theorem leads to two tangents of a circle from a point outside the circle, and their tangent lengths are equal. The straight line connecting a point outside the circle and the center of the circle bisects the angle between two tangents from that point to the circle.

50。 The degree of the chord tangent angle is equal to half the degree of the arc it encloses. The tangent angle is equal to the circumferential angle of the arc it encloses.

5 1。 Intersecting chord theorem; Cutting line theorem; secant theorem

10 1 A circle is a set of points whose distance from a fixed point is equal to a fixed length.

102 The interior of a circle can be regarded as a collection of points whose center distance is less than the radius.

The outer circle of 103 circle can be regarded as a collection of points whose center distance is greater than the radius.

104 The radius of the same circle or equal circle is the same.

The distance from 105 to the fixed point is equal to the trajectory of a fixed-length point, which is a circle with the fixed point as the center and the fixed length as the radius.

106 and the locus of the point with the same distance between the two endpoints of the known line segment is the middle vertical line of the line segment.

The locus from 107 to a point with equal distance on both sides of a known angle is the bisector of this angle.

The trajectory from 108 to the equidistant point of two parallel lines is a straight line parallel and equidistant to these two parallel lines.

Theorem 109 determines a straight line from three points that are not on a straight line.

1 10 vertical diameter theorem divides the chord perpendicular to its diameter into two parts, and divides the two arcs opposite to the chord into two parts.

1 1 1 inference 1 ① bisect the diameter of the chord (not the diameter) perpendicular to the chord and bisect the two arcs opposite the chord.

(2) The perpendicular line of the chord passes through the center of the circle and bisects the two arcs opposite to the chord.

③ bisect the diameter of an arc opposite to the chord, bisect the chord vertically, and bisect another arc opposite to the chord.

1 12 Inference 2 The arcs sandwiched by two parallel chords of a circle are equal.

1 13 circle is a centrosymmetric figure with the center of the circle as the symmetry center.

Theorem 1 14 In the same circle or in the same circle, the isocentric angle has equal arc, chord and chord center distance.

1 15 It is inferred that in the same circle or equal circle, if one set of quantities in two central angles, two arcs, two chords or the chord-center distance between two chords is equal, the corresponding other set of quantities is also equal.

Theorem 1 16 The angle of an arc is equal to half its central angle.

1 17 Inference 1 The circumferential angles of the same arc or the same arc are equal; In the same circle or in the same circle, the arcs of equal circumferential angles are also equal.

1 18 Inference 2 The circumferential angle (or diameter) of a semicircle is a right angle; A chord with a circumferential angle of 90 is a diameter.

1 19 Inference 3 If the median line of one side of a triangle is equal to half of this side, then this triangle is a right triangle.

120 Theorem The diagonals of the inscribed quadrilateral of a circle are complementary, and any external angle is equal to its internal angle.

12 1① the intersection of the straight line l and ⊙O is d < r.

(2) the tangent of the straight line l, and ⊙ o d = r.

③ lines l and ⊙O are separated by d > r.

122 tangent theorem The straight line passing through the outer end of the radius and perpendicular to the radius is the tangent of the circle.

123 The property theorem of tangent line The tangent line of a circle is perpendicular to the radius passing through the tangent point.

124 Inference 1 A straight line passing through the center of the circle and perpendicular to the tangent must pass through the tangent point.

125 Inference 2 A straight line passing through the tangent and perpendicular to the tangent must pass through the center of the circle.

The tangent length theorem 126 leads to two tangents of a circle from a point outside the circle, and their tangent lengths are equal. The line between the center of the circle and this point bisects the included angle of the two tangents.

127 The sum of two opposite sides of a circle's circumscribed quadrilateral is equal.

128 Chord Angle Theorem The chord angle is equal to the circumferential angle of the arc pair it clamps.

129 Inference: If the arc enclosed by two chord tangent angles is equal, then the two chord tangent angles are also equal.

130 intersection chord theorem The length of two intersecting chords in a circle divided by the product of the intersection point is equal.

1365438+

132 tangent theorem leads to the tangent and secant of a circle from a point outside the circle, and the tangent length is the middle term in the length ratio of the two lines at the intersection of this point and secant.

133 It is inferred that two secant lines of the circle are drawn from a point outside the circle, and the product of the lengths of the two lines from that point to the intersection of each secant line and the circle is equal.

134 If two circles are tangent, then the tangent point must be on the line.

135① perimeter of two circles D > R+R ② perimeter of two circles d = r+r.

③ the intersection of two circles r-r < d < r+r (r > r).

④ inscribed circle D = R-R (R > R) ⑤ two circles contain D < R-R (R > R).

Theorem 136 The intersection of two circles bisects the common chord of two circles vertically.

Theorem 137 divides a circle into n (n ≥ 3);

(1) The polygon obtained by connecting the points in turn is the inscribed regular N polygon of this circle.

(2) The tangent of a circle passing through each point, and the polygon whose vertex is the intersection of adjacent tangents is the circumscribed regular N polygon of the circle.

Theorem 138 Any regular polygon has a circumscribed circle and an inscribed circle, which are concentric circles.

139 every inner angle of a regular n-polygon is equal to (n-2) ×180/n.

140 Theorem Radius and apothem Divides a regular N-polygon into 2n congruent right triangles.

14 1 the area of the regular n polygon Sn = PNRN/2 P represents the perimeter of the regular n polygon.

142 The area of a regular triangle √ 3a/4a indicates the side length.

143 if there are k positive n corners around a vertex, since the sum of these angles should be 360, then K× (n-2) 180/n = 360 becomes (n-2)(k-2)=4.

144 arc length calculation formula: L = n ∏ R/ 180.

145 sector area formula: s sector = n ∏ R/360 = LR/2 146 inner common tangent length = d-(R-r) outer common tangent length = d-(R+r).

High school geometry

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 non-planar.

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.

Positional relationship between straight line and plane: There are only three positional relationships between straight line and plane: in the plane, intersecting with 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 line and plane: 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, we can get that the projection of the vertex on the bottom is the vertical center of the triangle on the bottom through the three perpendicular theorem.

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.

note:

1, pay attention to the establishment of spatial rectangular coordinate system.

2. Space vectors can be applied without coordinate system.

Euler formula of polyhedron: v (angle) +F (surface) -E (edge) =2.

There are only five regular polyhedrons: regular four, regular six, regular eight, regular twelve and regular icosahedron.

ball

note:

1, the difference between ball and ball area

2. Longitude (plane angle) and latitude (line plane angle)

3, the surface area and volume formula of the ball

4. The multiple of the distance between two parallel planes on the sphere.