Overall summary of mathematical formula theorem in the second day of junior high school

Give you a complete set of junior high school geometry knowledge.

1 There is only one straight line at two points.

The line segment between two points is the shortest.

The complementary angles of the same angle or equal angle are equal.

The complementary angles of the same angle or the same angle are equal.

One and only one straight line is perpendicular to the known straight line.

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

7 Parallelism axiom: After passing a point outside a straight line, there is one and only one straight line parallel to this straight line:

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

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

The internal dislocation angles of 10 are equal, and the two straight lines are parallel.

1 1 are complementary and two straight lines are parallel.

12 Two straight lines are parallel and have the same angle.

13 two straight lines are parallel, and the internal dislocation angles are equal.

14 Two straight lines are parallel and complementary.

Theorem 15: The sum of two sides of a triangle is greater than the third side.

16 inference: the difference between two sides of a triangle is smaller than the third side.

17 interior angle sum theorem: the sum of three interior angles of a triangle is equal to 180.

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

19 Inference 2: One outer angle of a triangle is equal to the sum of two non-adjacent inner angles.

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

2 1 congruent triangles has equal sides and angles.

Angular axiom: There are two congruent triangles (SAS) with two sides and their included angles are equal.

Axiom of 23 angles: Two triangles (ASA) with two angles and congruent sides.

Inference: There are two angles, and the opposite side of one angle corresponds to the congruence (AAS) of two triangles.

Axiom of 25 sides: congruence (SSS) of two triangles corresponding to three sides.

Axiom of hypotenuse and right-angled edge: The hypotenuse and right-angled edge of two right-angled triangles correspond to equal congruence (HL).

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

Theorem 2: In an angle, points with equal distance to both sides of the angle are on the bisector of the angle.

The bisector of an angle 29 is the set of all points with equal distance to both sides of the angle.

The property theorem of isosceles triangle: the two base angles of isosceles triangle are equal.

3 1 inference 1: the bisector of the top angle of an isosceles triangle, which bisects and is perpendicular to the bottom.

The bisector of the top corner, the midline of the bottom edge and the height of the isosceles triangle coincide.

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.

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

The center line of the hypotenuse of a right triangle is equal to half of the hypotenuse.

Theorem 39: The distance between the point on the vertical line of a line segment and the two endpoints of this line segment is equal.

40 Inverse Theorem: The point where the two endpoints of a line segment are at the same distance is on the middle vertical line of this line segment.

The perpendicular bisector of the 4 1 line segment can be regarded as the set of all points with equal distance from 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 vertically bisected by the same straight line, then the two graphs are symmetrical about this straight line.

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, a2+b2=c2.

47 Pythagorean Theorem Inverse Theorem: If the lengths of three sides of triangle A, B and C satisfy the relationship a2+b2=c2, then this triangle is a right triangle, and ∠C=900.

Theorem 48: The sum of the internal angles of a 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 50 polygons: the sum of internal angles of n polygons is equal to (n-2) × 180.

5 1 inference: the sum of the external angles of any polygon is equal to 360.

52 parallelogram property theorem 1: the two diagonal lines of parallelogram are equal respectively.

53 parallelogram property theorem 2: two groups of opposite sides of parallelogram are equal respectively.

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

55 parallelogram property theorem 3: the two diagonals of parallelogram are equally divided.

56 parallelogram decision theorem 1: two groups of quadrangles with equal diagonals are parallelograms.

57 parallelogram decision theorem 2: two groups of quadrilaterals with equal opposite sides are parallelograms.

58 parallelogram Decision Theorem 3: A quadrilateral whose diagonal is bisected is a parallelogram.

59 parallelogram decision theorem 4: A group of parallelograms with parallel and equal opposite sides are parallelograms.

60 Rectangular Property Theorem 1: All four corners of a rectangle are right angles.

6 1 rectangle property theorem 2: the diagonals of rectangles are equal.

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

63 Rectangular Decision Theorem 2: Parallelograms with equal diagonals are rectangles.

64 diamond property theorem 1: all four sides of a diamond are equal.

Theorem 2: Diagonal lines of rhombus are perpendicular to each other, and each diagonal line bisects a set of diagonal lines.

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

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

68 Diamond Decision Theorem 2: Parallelograms whose diagonals are perpendicular to each other are diamonds.

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

Theorem 2: Two diagonal lines of a square are equal and bisected vertically, and each diagonal line bisects a set of diagonal lines.

Theorem 7 1 1: Two centrally symmetric graphs are congruent.

Theorem 2: For 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.

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

Theorem 2: The two diagonals of an isosceles trapezoid are equal.

Judgment theorem of isosceles trapezoid: two trapezoid with equal angles on the same base are isosceles trapezoid.

A trapezoid with equal diagonal lines is an isosceles trapezoid.

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

79 Inference 1: A straight line passing through the midpoint of one waist of a trapezoid and parallel to the two bottoms 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.

8 1 midline theorem of triangle: the midline of triangle is parallel to the third side and equal to half of it.

Mean value theorem of trapezium: the median line of trapezium is parallel to the two bottoms and equal to half of the sum of the two bottoms L=(a+b)÷2, S = L× H.

Basic properties of 83 ratio (1): If a:b=c:d, then ad=bc.

(2) if ad=bc, (a, b, c, d≠0), then a: b = c: d.

Proportional nature: if a:b=c:d, then (a b): b = (c d): d.

85 Proportional property: If a:b=c:d=…=m:n(b+d+…+n≠0), then (a+c+…+m): (b+d+…+n) = a: b.

Proportional theorem of parallel lines and line segments: three parallel lines cut two straight lines, and the corresponding line segments are proportional.

Inference: A straight line parallel to one side of a triangle intersects with the other two sides (or extension lines of both sides), and the corresponding line segments are proportional.

Theorem 88: If the corresponding line segments cut by two sides of a triangle (or the extension lines of two sides) are proportional, the straight line is parallel to the third side of the triangle.

A straight line parallel to one side of a triangle and intersecting with the other two sides, the three sides of the cut triangle are directly proportional to the three sides of the original triangle.

Theorem 90: A straight line parallel to one side of a triangle intersects with the other two sides (or extension lines of both sides), and the triangle formed is similar to the original triangle.

9 1 similar triangles's Judgment Theorem 1: Two angles correspond equally and two triangles are similar.

Two right-angled triangles divided by the height on the hypotenuse are similar to the original triangle (photography theorem)

Decision Theorem 2: Two triangles with equal included angles and corresponding proportional sides are similar.

Decision Theorem 3: Similarity of three sides corresponding to two proportional triangles.

Theorem 95: If the hypotenuse and a right-angled side of a right-angled triangle are proportional to the hypotenuse and a right-angled side of another right-angled triangle, then the two right-angled triangles are similar.

96 Property Theorem 1: The ratio of the corresponding height, the ratio of the corresponding median line and the ratio of the corresponding angular bisector of similar triangles are all equal to the similarity ratio.

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

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

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.

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

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 distance from the center of the circle is less than the radius.

103 The outer circle of a circle can be regarded as a collection of points whose distance from the center of the circle 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.

The locus of the point where 106 is equal to the distance between the two endpoints of a known line segment is the midline of this 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: Three points that are not on a straight line determine a circle.

1 10 vertical diameter theorem: the diameter perpendicular to the chord bisects the chord and bisects the two arcs opposite the chord.

1 1 1 inference 1: ① bisect the diameter of the chord perpendicular to the chord (not the diameter) and bisect the two arcs opposite to 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, equal central angles have equal arcs, equal chords and equal chord center distances.

1 15 Inference: In the same circle or equal circle, if one of two central angles, two arcs, two chords or the chord-to-chord distance of two chords is equal, the corresponding other group is also equal.

Theorem 1 16: The angle of an arc to a circle is equal to half its angle to the center of the circle.

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

1 18 corollary 2: the circumferential angle of a semicircle (or diameter) 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.

Theorem 120: Diagonal lines of inscribed quadrangles of a circle are complementary, and any external angle is equal to its internal angle.

12 1① line l intersects with ⊙O D R

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

③ straight 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 property theorem of tangent: the tangent 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 that crosses the tangent point and is perpendicular to the tangent line must pass through the center of the circle.

126 tangent length theorem: two tangents drawn 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.

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

128 chord tangent angle theorem: the chord tangent 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.

13 1 inference: If the chord intersects the diameter vertically, then half of the chord is the proportional median of the two line segments formed by dividing it by the diameter.

132 Secant Theorem: The tangent and secant of a circle are drawn from a point outside the circle, and the length of the tangent is the middle term of the ratio of the lengths of the two lines from this point to the intersection of the secant and the circle.

133 inference: the product of two secant lines of a circle from a point outside the circle to the intersection of each secant line and the circle is equal.

134 If two circles are tangent, then the connecting line must cross the tangent point.

135① the distance between two circles is d+r+r.

(2) circumscribed circle d d = r+r.

③ Two circles intersect 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 the two circles vertically.

Theorem 137: Divide a circle into n equal parts (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.

Theorem 140: The radius and vertex of a regular N-polygon divide the 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 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.