Point-line angle theorem;

Theorem of points: there is only one straight line passing through two points.

Point theorem: the shortest line segment between two points.

Angle theorem: the complementary angles of the same angle or equal angle are equal.

Angle theorem: the complementary angles of the same angle or equal angle are equal.

Straight line theorem: there is only one straight line perpendicular to the known straight line at one point.

Straight line theorem: Of all the line segments connecting a point outside a straight line with a point on a straight line, the vertical line segment is the shortest.

Parallelism theorem:

After passing a point outside the straight line, there is one and only one straight line parallel to this straight line.

Inference: If both lines are parallel to the third line, then the two lines are also parallel to each other.

Parallel nature:

1, equal to the complementary angle, and two straight lines are parallel.

2. The internal dislocation angles are equal and the two straight lines are parallel.

3. The internal angles on the same side are complementary and the two straight lines are parallel.

Parallel reasoning:

1, two straight lines are parallel and have the same angle.

2. The two straight lines are parallel and the internal dislocation angles are equal.

These two straight lines are parallel and complementary.

Triangle interior angle theorem;

Theorem: The sum of two sides of a triangle is greater than the third side.

Inference: The difference between two sides of a triangle is smaller than the third side.

Theorem of sum of interior angles of triangle: the sum of three interior angles of triangle is equal to 180.

Inference 1: The two acute angles of a right triangle are complementary.

Inference 2: One external angle of a triangle is equal to the sum of two non-adjacent internal angles.

Inference 3: An outer angle of a triangle is larger than any inner angle that is not adjacent to it.

Congruent triangles's judgment theorem;

Theorem: The sides and angles corresponding to congruent triangles are equal.

Edge Theorem (SAS): Two triangles have two sides, and their included angles are congruent.

Angle Theorem (ASA): Two triangles have two angles and their sides are congruent.

Inference (AAS): Two triangles with two angles and opposite sides of one angle are congruent.

Edge Theorem (SSS): Two triangles corresponding to three equilateral sides are congruent.

Hypotenuse and right-angled edge theorem (HL): Two right-angled triangles with hypotenuse and a right-angled edge are congruent.

Angular bisector theorem;

Theorem 1: The distance from a point on the bisector of an angle to both sides of the angle is equal.

Theorem 2: The points with equal distance on both sides of an angle are on the bisector of this angle.

The bisector of an angle is the set of all points with equal distance to both sides of the angle.

Property theorem of isosceles triangle;

The two base angles of an isosceles triangle are equal (that is, equilateral and equilateral).

Inference 1: The bisector of the vertex of an isosceles triangle bisects the base and is perpendicular to the base.

The bisector of the top angle, the median line on the bottom edge and the height on the bottom edge of the isosceles triangle coincide with each other.

Inference 3: All angles of an equilateral triangle are equal, and each angle is equal to 60.

Judgment theorem of isosceles triangle: If the two angles of a triangle are equal, then the opposite sides of the two angles are also equal (equal angles and equal sides).

Inference 1: A triangle with three equal angles is an equilateral triangle.

Inference 2: An isosceles triangle with an angle equal to 60 is an equilateral triangle.

Symmetry theorem

Theorem: The distance between the point on the vertical line of a line segment and the two endpoints of this line segment is equal.

Inverse theorem: the point where the two endpoints of a line segment are equidistant is on the middle vertical line of this line segment.

The middle vertical line of a line segment can be regarded as a set of all points with the same distance at both ends of the line segment.

Theorem 1: congruence of two graphs symmetric about a straight line.

Theorem 2: If two figures are symmetrical about a straight line, then the symmetry axis is the perpendicular line connecting the corresponding points.

Theorem 3: Two figures are symmetrical about a straight line. If their corresponding line segments or extension lines intersect, then the intersection point is on the axis of symmetry.

Inverse theorem: If the straight line connecting the corresponding points of two graphs is bisected vertically by the same straight line, then the two graphs are symmetrical about this straight line.

Right triangle theorem:

Theorem: In a right triangle, if an acute angle is equal to 30, the right side it faces is equal to half of the hypotenuse.

Decision theorem: the median line on the hypotenuse of a right triangle is equal to half of the hypotenuse.

Pythagorean theorem: the sum of squares of two right angles A and B of a right triangle is equal to the square of hypotenuse C, that is, A? +b？ =c？ .

Inverse theorem of Pythagorean theorem: If three sides of a triangle are related to A, B, C? +b？ =c？ Then this triangle is a right triangle.

Theorem of sum of interior angles of polygons;

Theorem: The sum of internal angles of quadrilateral is equal to 360; The sum of the external angles of the quadrilateral is equal to 360.

Theorem of the sum of internal angles of polygons: the sum of internal angles of n polygons is equal to (n-2) × 180.

Inference: The sum of the external angles of any polygon is equal to 360.

Parallelogram theorem:

The property theorem of parallelogram 1: the diagonals of parallelogram are equal

2. The opposite sides of the parallelogram are equal.

3. The diagonal bisection of parallelogram.

Inference: The parallel segments sandwiched between two parallel lines are equal.

Parallelogram decision theorem 1: Two sets of quadrilaterals with equal diagonals are parallelograms.

2. Two groups of quadrilaterals with equal opposite sides are parallelograms.

3. Quadrilaterals whose diagonals are bisected with each other are parallelograms.

4. A set of parallelograms with equal opposite sides is a parallelogram.

Rectangular theorem

Attribute: 1: All four corners of a rectangle are right angles.

2: The diagonals of the rectangles are equal.

Decision: 1: A quadrilateral with three right angles is a rectangle.

2. Parallelograms with equal diagonals are rectangles.

diamond

1: All four sides of the diamond are equal.

2. The diagonals of the diamond are perpendicular to each other, and each diagonal bisects a set of diagonals.

Diamond area = half of diagonal product, that is, S=(a×b)÷2.

Diamond decision theorem

1: A quadrilateral with four equilateral sides is a diamond.

2. Parallelograms with diagonal lines perpendicular to each other are diamonds.

Square theorem:

Theorem of Square Properties 1: All four corners of a square are right angles and all four sides are equal.

2. The two diagonals of a square are equal and equally divided vertically, and each diagonal is equally divided into a set of diagonals.

Central symmetry theorem;

Theorem 1: congruence of two graphs with central symmetry.

2. Regarding two graphs with symmetrical centers, the connecting lines of symmetrical points pass through the symmetrical center and are equally divided by the symmetrical center.

Inverse theorem: If a straight line connecting the corresponding points of two graphs passes through a point and is bisected by the point, then the two graphs are symmetrical about the point.

Theorem of isosceles trapezoid properties;

Property theorem of isosceles trapezoid: 1. The two angles of an isosceles trapezoid on the same base are equal.

2. The two diagonals of the isosceles trapezoid are equal.

Judgment theorem of isosceles trapezoid: 1. A trapezoid with two equal angles on the same base is an isosceles trapezoid.

2. A trapezoid with equal diagonal lines is an isosceles trapezoid.

Theorem of bisection of parallel lines: If a group of parallel lines have equal segments on a straight line, then the segments on other straight lines are also equal.

Inference 1: A straight line passing through the midpoint of one waist of a trapezoid and parallel to the bottom will bisect the other waist.

Inference 2: A straight line passing through the midpoint of one side of a triangle and parallel to the other side will bisect the third side.

pappus law

Triangle: The center line of a triangle is parallel to the third side and equal to half of it.

Trapezoid: the midline of the trapezoid is parallel to the two bottoms and equal to half of the sum of the two bottoms. L = (a+b) ÷ 2s = l× h.

Similar triangles theorem;

A straight line parallel to one side of a triangle intersects with the other two sides (or extension lines of both sides) to form a triangle similar to the original triangle.

Similar triangles's Judgment Theorem 1: Two angles are equal and two triangles are similar (ASA).

Two right triangles divided by the height on the hypotenuse are similar to the original triangle.

2: The two sides are proportional and the included angle is equal, and the two triangles are similar (SAS).

3: Three sides are proportional and two triangles are similar (SSS)

Theorem of Similar Right Triangle: If the hypotenuse and right side of one right triangle are proportional to the hypotenuse and right side of another right triangle, then the two right triangles are similar.

Similar nature:

1: similar triangles corresponds to the height ratio, the ratio corresponding to the center line and the ratio corresponding to the angular bisector are all equal to the similarity ratio.

2. The ratio of similar triangles perimeter is equal to similarity ratio.

3. The ratio of similar triangles area is equal to the square of similarity ratio.

Trigonometric function theorem;

The sine of any acute angle is equal to the cosine of the remaining angles, and the cosine of any acute angle is equal to the sine of the remaining angles.

The tangent of any acute angle is equal to the cotangent of the remaining angles, and the cotangent of any acute angle is equal to the tangent of the remaining angles.

Theorem of circle;

1. A circle is determined by three points of the line * * *. Countless circles can be made by passing through one point, and countless circles can be made by passing through two points, and the center of the circle is on the perpendicular to the line segment connecting these two points.

Theorem: A three-point crossing line can be made into one and only one circle.

Inference: The perpendicular lines of the three sides of a triangle intersect at a point, which is the outer center of the triangle.

The intersection of three high lines of a triangle is called the vertical center of the triangle.

2. Vertical diameter theorem

A circle is a figure with central symmetry; The center of the circle is its symmetrical center, the circle is a circumferentially symmetrical figure, and any straight line passing through the center of the circle is its symmetrical axis.

Theorem: bisect the chord perpendicular to the diameter of the chord and score the two arcs subtended by the chord.

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

Inference 2: The perpendicular bisector of a chord passes through the center of the circle and bisects the two arcs opposite to the chord.

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

3. Distance between arc, chord and chord center

Theorem: In the same circle or in the same circle, the chords of equal arcs are equal, and the distance between the chords' centers is equal.

4. The positional relationship between a circle and a straight line

If a straight line and a circle have nothing in common, we say that the straight line and the circle are separated.

If a straight line and a circle have only one common point, we say that the straight line is tangent to the circle. This straight line is called the tangent of the circle, and this common point is called their tangent point.

Theorem: A straight line passing through the outer radius of a circle and perpendicular to this radius is the tangent of this circle.

Theorem: The tangent of a circle passes vertically through the radius of the tangent point.

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

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

If a straight line and a circle have two common points, we say that the straight line and the circle intersect, which is called the secant of the circle, and these two common points are called their intersection.

The positional relationship between a straight line and a circle can only be divided into three types: separation, tangency and intersection.

5. The inscribed circle of a triangle

If the straight line of each side of a polygon is tangent to a circle, the polygon is called the circumscribed polygon of the circle and the circle is called the inscribed circle of the polygon.

Theorem: The bisectors of three internal angles of a triangle intersect at one point, which is the center of the triangle.

6. Tangent length theorem

Theorem: Two tangents leading from a point outside the circle are equal in length, and the connecting line between the center of the circle and this point bisects the included angle of the two tangents.

7. The circumscribed quadrilateral of a circle

Theorem: The sum of two opposite sides of the circumscribed quadrangle of a circle is equal.

Theorem: If the sum of two opposite sides of a quadrilateral is equal, then it must have an inscribed circle.

8. The positional relationship between two circles

On the plane, the positional relationship between two non-overlapping circles has the following five situations: external separation, external tangency, intersection, internal tangency and external tangency.

A straight line passing through the centers of two circles is called the connecting line of two circles, and the distance between the centers is called the center distance.

Theorem: The connecting line of two circles is the symmetry axis of two circles. When two circles are tangent, their tangents are all on the connecting line.

(1) The outer distance between two circles is d >；; R+r (2) circumscribed circle D = R+R.

(3) The intersection of two circles R-rr) (4) The inscribed circle D = R-R (R > r)

(5) two circles contain dr)

In special cases, two circles are concentric circles, and d=0.

9. The common tangent of two circles

Theorem: the appearance of two tangents of two circles, etc. The lengths of the two internal common tangents of two circles are also equal.

Theorem of proportional properties;

Basic properties of (1) ratio

If A: B = C: D, then ad=bc If ad=bc, then A: B = C: D.

(2) Combination characteristics

If a/b=c/d, then (a b)/b = (c d)/d.

(3) Equidistant property

If a/b=c/d=…=m/n(b+d+…+n≠0), then (a+c+…+m)/(b+d+…+n) = a/b.

Basic formula of mathematics in senior high school entrance examination

Formulas for circles and arcs:

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

Arc length calculation formula: L=n R/ 180.

Sector area formula: s sector =n r 2/360 = lr/2.

(1) Two circles are separated by d>, R+R2 is circumscribed by two circles, D = R+R3 is intersected by two circles, R-R+R2) ④ is inscribed by two circles, D = R-R (R > R) (5) Two circles contain dr)

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

Theorem: Divide the circle into n (n ≥ 3): ⑴ The polygon obtained by connecting all points in turn is an inscribed regular N polygon of the circle ⑴ The circle passes through the tangents of all points, and the polygon whose vertices are the intersections of adjacent tangents is an circumscribed regular N polygon of the circle.

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

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.

Factorization formula:

Formula: A 3+B 3+C 3-3 ABC = (A+B+C) (A? +b？ +c？ -ab-bc-ca)

Solution: A 3+B 3+C 3-3 ABC

=(a+b)(a^2-ab+b^2)+c(c^2-3ab)

=(a+b)(a^2-ab+b^2)+c(c^2-3ab+a^2-ab+b^2-a^2+ab-b^2)

=(a+b)(a^2-ab+b^2)+c[(c^2-a^2-2ab-b^2)+(a^2-ab+b^2)]

=(a+b)(a^2-ab+b^2)+c[c^2-(a+b)^2]+c(a^2-ab+b^2)

=(a+b+c)(a^2-ab+b^2)+c(a+b+c)(c-a-b)

=(a+b+c)(a^2+b^2+c^2-ab-bc-ac)

Square difference formula: a square -b square =(a+b)(a-b)

Complete sum of squares formula: (a+b) square =a? +2ab+b？

Complete square difference formula: (a-b) square =a? -2ab+b？

Two kinds: ax? +bx+c=a[x-(-b+√(b？ -4ac))/2a][x-(-b-√(b？ -4ac))/2a] double root type

Cubic sum formula: a 3+b 3 = (a+b) (a? -ab+b？ )

Cubic difference formula: a 3-b 3 = (a-b) (a? +ab+b？ )

Complete cubic formula: a^3 3a? b+3ab？ b^3=(a b)^3.

The formula and discriminant of quadratic equation in one variable;

The solution of quadratic equation in one variable-b+√ (b? -4ac)/2a，-b-√(b？ -4ac)/2a

The relationship between root and coefficient x1+x2 =-b/ax1* x2 = c/a Note: Vieta theorem.

discriminant

b？ -4ac=0 Note: The equation has two equal real roots.

b？ -4ac & gt； 0 Note: The equation has two unequal real roots.

b？ -4ac & lt； Note: The equation has no real root, but a complex number of the yoke.

Trigonometric inequality:

|a+b|≤|a|+|b| |a-b|≤|a|+|b|

| a |≤b-b≤a≤b | a-b |≥| a |-| b |-| a |≤a ≤| a |

Arithmetic series formula:

The sum of the first n terms of some series:

1+2+3+4+5+6+7+8+9+…+n = n(n+ 1)/2 1+3+5+7+9+ 1 1+ 13+ 15+…+(2n- 1)= N2

2+4+6+8+ 10+ 12+ 14+…+(2n)= n(n+ 1) 12+22+32+42+52+62+72+82+…+N2 = n(n+ 1)(2n+ 1)/6

13+23+33+43+53+63+…n3 = N2(n+ 1)2/4 1 * 2+2 * 3+3 * 4+4 * 5+5 * 6+6 * 7+…+n(n+ 1)= n(n+ 1)(n+2)/3

Formulas of trigonometric functions-the sum formula of two angles;

sin(A+B)=sinAcosB+cosAsinB

sin(A-B)=sinAcosB-sinBcosA

cos(A+B)=cosAcosB-sinAsinB

cos(A-B)=cosAcosB+sinAsinB

tan(A+B)=(tanA+tanB)/( 1-tanA tanB)

tan(A-B)=(tanA-tanB)/( 1+tanA tanB)

Formulas of trigonometric functions-Double Angle Formula:

tan2A=2tanA/( 1-tan2A)

cos2a=cos？ Sin. a=2cos？ a- 1= 1-2sin？ a

Formulas of trigonometric functions-Half Angle Formula:

sin(A/2)=√(( 1-cosA)/2)sin(A/2)=-√(( 1-cosA)/2)

cos(A/2)=√(( 1+cosA)/2)cos(A/2)=-√(( 1+cosA)/2)

tan(A/2)=√(( 1-cosA)/(( 1+cosA))tan(A/2)=-√(( 1-cosA)/(( 1+cosA))

Formulas of trigonometric functions-sum-difference product:

2 Sina cosb = sin(A+B)+sin(A-B)2 cosa sinb = sin(A+B)-sin(A-B)

2 cosa cosb = cos(A+B)-sin(A-B)2 Sina sinb = cos(A+B)-cos(A-B)

sinA+sinB = 2 sin((A+B)/2)cos(A-B)/2 cosA+cosB = 2 cos((A+B)/2)sin((A-B)/2)

tanA+tanB = sin(A+B)/cosa cosb tanA-tanB = sin(A-B)/cosa cosb

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