The complete set of junior high school mathematics knowledge points 1, the condition of the root of a linear equation is △=b2-4ac. When △ > 0, the quadratic equation has two unequal real roots; When △=0, the unary quadratic equation has two identical real roots; When △ < 0, the quadratic equation of one variable has no real root 2. The properties of parallelogram are as follows: ① Two groups of parallelograms with parallel opposite sides are called parallelograms. (2) The line segment connected by two nonadjacent vertices of a parallelogram is called its diagonal. ③ The opposite sides/diagonals of parallelogram are equal. (4) The diagonal of the parallelogram is equally divided. Diamond: ① A group of parallelograms with equal adjacent sides is a diamond; ② The four sides of the collar are equal, and the two diagonal lines are equally divided vertically, and each diagonal line is equally divided into a set of diagonal lines. ③ Judgment conditions: define a parallelogram with vertical diagonal and a quadrilateral with equal sides. Rectangular and square: ① A parallelogram with a right angle is called a rectangle. ② The diagonals of a rectangle are equal, and all four corners are right angles. ③ Parallelograms with equal diagonals are rectangles. ④ A square has all the properties of parallelogram, rectangle and diamond. ⑤ A set of rectangles with equal adjacent sides is a square. Polygon: ① The sum of the internal angles of n polygons is equal to (N-2) 180 degrees ② The angle formed by one side of the internal angle of a polygon and the opposite extension line of the other side is called the external angle of this polygon. Take an outer angle of the polygon at each vertex, and the sum of them is called the average sum of the inner angles of the polygon (both equal to 360 degrees): for n numbers, X 1, X2…XN, we call (X 1+X2+…+XN)/N the arithmetic average of the n numbers, and record it as the weighted average of x: a set of data. Every data is often given a weight, which is the weight 2, the basic theorem 1, only one straight line when crossing two points 2, the shortest line segment between two points 3, the same angle or the same angle's complementary angle 4, the same angle or the same angle's complementary angle 5, only one straight line perpendicular to the known straight line when crossing a point 6, among all the line segments connected with points on the straight line, the vertical line segment is the shortest, and 7, the parallel axiom passes outside the straight line. 8. If two straight lines are parallel to the third straight line, the two straight lines are also parallel to each other. 9. The congruence angle is equal, two straight lines are parallel to 10, the internal dislocation angle is equal, two straight lines are parallel to 1 1, the internal angles on the same side are complementary, and two straight lines are parallel to 12, and the congruence angle is equal. The offset angle is equal to 14, and the two straight lines are parallel. The inner angles of the same side are complementary to each other 15, the sum of the two sides of the theorem triangle is greater than the third side 16, the difference between the two sides of the inference triangle is less than the third side 17, the sum of the inner angles of the triangle and the three inner angles of the theorem triangle is equal to 180 18, and the two acute angles of the inference triangle are mutual. Inference 2: One external angle of a triangle is equal to the sum of two non-adjacent internal angles; Inference 3: One outer angle of a triangle is larger than any inner angle that is not adjacent to it; 2 1; The corresponding edge of congruent triangles is equal to the corresponding angle; 22; Angular axiom (SAS) has two triangles with equal angles; 23; Corner axiom (ASA). The congruence of two triangles with two angles corresponding to their clamping sides is 24, the congruence of two triangles with two angles corresponding to the opposite sides of an angle is 25 in inference, the congruence of two triangles with three sides is 26 in the side-by-side axiom, and the congruence of right triangles with two hypotenuses and a right-angled side is 27 in the hypotenuse and right-angled side axiom (HL). Theorem 1 bisector of an angle. The bisector of an angle is the set of all points with equal distance to both sides of the angle 30. The property theorem of isosceles triangle: the two base angles of isosceles triangle are equal (that is, equal corners) 3 1. It is inferred that the bisector of the top angle of the isosceles triangle bisects the bottom surface and is perpendicular to the bottom surface 32, and the bisector of the top angle of the isosceles triangle, the midline on the bottom surface and the height on the bottom surface coincide. Inference 3 All angles of an equilateral triangle are equal, and each angle is equal to 60 34. Decision Theorem of Isosceles Triangle If a triangle has two equal angles, then the opposite sides of the two angles are equal (equal sides) 35. Inference 1 A triangle with three equal angles is an equilateral triangle 36. Inference 2 An isosceles triangle with an angle equal to 60 is an equilateral triangle 37. If the acute angle is equal to 30, the right-angled side it faces is equal to half of the hypotenuse 38, the median line on the hypotenuse of the right triangle is equal to half of the hypotenuse 39, the distance between the point on the perpendicular line of the theorem line segment and the two endpoints of the line segment is equal to 40, and the inverse theorem and the point with the same distance between the two endpoints of a line segment are 4 1 on the perpendicular line of the line segment. The middle vertical line of the line segment can be regarded as the set of all points with equal distance from both ends of the line segment. Theorem 1 Two figures symmetrical about a straight line are conformal. Theorem 2 If two figures are symmetrical about a straight line, the symmetry axis is perpendicular bisector 44 connecting the corresponding points. Theorem 3 Two figures are symmetrical about a straight line. If their corresponding line segments or extension lines intersect, the intersection point is on the symmetry axis. 45. Inverse Theorem If the connecting line of the corresponding points of two graphs is vertically bisected by the same straight line, then the two graphs are symmetrical about this straight line. 46. Pythagorean Theorem The sum of squares of two right-angled sides A and B of a right triangle is equal to the square of hypotenuse C, that is, a2+b2=c2 47 and the inverse theorem of Pythagorean Theorem. If the three sides of a triangle are related to A, B and C, and a2+b2=c2, then this triangle is a right triangle 48. The sum of the inner angles of quadrilateral is equal to 360 49, the sum of the outer angles of quadrilateral is equal to 360 50, and the sum of the inner angles of polygon and theorem N is equal to (n-2) ×1805/kloc-. It is inferred that the parallel line segment sandwiched between two parallel lines is equal to 55. The parallelogram property theorem 3 The diagonal of the parallelogram is bisected by 56. Parallelogram decision theorem 1 Two sets of quadrilaterals with equal diagonals are parallelograms 57. Parallelogram Decision Theorem 2 Two groups of quadrangles with equal opposite sides are parallelograms 58. Parallelogram Decision Theorem 3 The quadrangles whose diagonals are bisected are parallelogram 59 and parallelogram Decision Theorem 4. A set of parallelograms with parallel and equal opposite sides is parallelogram 60, rectangle property theorem 1, rectangle property theorem 2, rectangle diagonal equality 62, rectangle judgment theorem 1, and a quadrilateral with three right angles is rectangle 63. Rectangular Decision Theorem 2 The parallelogram with equal diagonals is a rectangle 64, the rhombus property theorem 1, the four sides of the rhombus are all equal 65, the rhombus property theorem 2, the diagonals of the rhombus are perpendicular to each other, each diagonal bisects a set of diagonals 66, and the rhombus area = half of the diagonal product. That is, S=(a×b)÷2 67, rhombus decision theorem 1, quadrilateral with four equal sides is rhombus 68, rhombus decision theorem 2, parallelogram with diagonal lines perpendicular to each other is rhombus 69, square property theorem 1, four corners of a square are right angles, four sides are equal 70, square property theorem 2, a square. Each diagonal bisects a set of diagonal lines 7 1. Theorem 1 is congruent with respect to two figures symmetrical with respect to the center 72. Theorem 2 is symmetrical with respect to two figures symmetrical with respect to the center. The connecting lines of symmetrical points all pass through the center of symmetry and are bisected by the center of symmetry 73. Inverse Theorem If the connecting lines of the corresponding points of two graphs pass through a certain point and are bisected by the point, then the two graphs are symmetrical about the point 74, isosceles trapezoid property theorem 75, isosceles trapezoid with two equal angles on the same base 76, isosceles trapezoid with two equal angles on the same base 77, isosceles trapezoid with two equal angles on the same base 78, and parallel line bisection theorem. If a group of parallel lines have equal segments on a straight line, then the segments cut on other straight lines are also equal. 79. Inference 1 Through a straight line parallel to the bottom of the trapezoid, the other waist 80 must be equally divided. Inference 2 Through a straight line parallel to the other side of a triangle, the third side must be equally divided. The midline theorem of a triangle is parallel to the third side. And equal to half of it. 82. The trapezoid midline theorem is parallel to the two bottoms and is equal to half of the sum of the two bottoms. L=(a+b)÷2 S=L×h 83, (1) Basic properties of the ratio: if a:b=c:d, then ad=bc, then a:b. Then (a b)/b = (c d)/d 85. The corresponding line segment obtained is proportional to 87. It is inferred that the straight line parallel to one side of the triangle cuts the other two sides (or the extension lines of both sides), and the corresponding line segment is proportional to 88. Theorem If the corresponding line segments obtained by cutting two sides (or extension lines of two sides) of a triangle are proportional, then this straight line is parallel to the third side 89 of the triangle, parallel to one side of the triangle and intersects with the other two sides. The three sides of the cut triangle are proportional to the three sides of the original triangle. 90. A straight line parallel to one side of a triangle intersects with the other two sides (or extension lines on both sides), and the triangle formed is similar to the original triangle 9 1, and the two angles of the similar triangle judgment theorem 1 are equal. Two triangles are similar (ASA) 92, two right triangles divided by the height on the hypotenuse are similar to the original triangle 93, decision theorem 2, two sides are proportional and the included angle is equal, two triangles are similar (SAS) 94, decision theorem 3, three sides are proportional and two triangles are similar (SSS) 95. Theorem 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 these two right-angled triangles are similar. Property theorem 1 similar triangles corresponding height ratio. The ratio of the corresponding median line to the bisector of the corresponding angle is equal to similarity ratio 97, property theorem 2, similar triangles perimeter ratio 98, property theorem 3, similar triangles area ratio 99, sine value of any acute angle is equal to cosine value of other angles, cosine value of any acute angle is equal to sine value of other angles 100, tangent value of any acute angle is equal to cotangent value of other angles, and cotangent value of any acute angle is equal to tangent value of other angles/kloc-0. Kloc-0/, a point set with a distance to a fixed point equal to fixed length 102, a point set with a radius of 103 and a point set with a radius of 104. The locus of a point with the same circle or the same circle radius equal to 105 and the distance from a fixed point equal to a fixed length is the locus of a circle with the fixed point as the center and the fixed length as the radius, the locus of a point with the same distance from two endpoints of a known line segment, and the locus of a point with the same distance from the middle perpendicular of the line segment 107 to both sides of a known angle. It is the bisector of this angle 108, the trajectory to the equidistant points of two parallel lines, and the straight line parallel to these two equidistant parallel lines 109, which proves that three points that are not on the same straight line determine a circle. 1 10, the vertical diameter theorem bisects the chord perpendicular to its diameter and bisects the two arcs opposite to the chord11,and it is inferred that 1 ① bisects the diameter (not the diameter) of the chord perpendicular to the chord, and bisects the median perpendicular of the two arcs opposite to the chord passing through the center. If the chord is divided vertically, the other arc bisected by the chord is 1 12. Inference 2 The arc between two parallel chords of a circle is equal to 1 13. A circle is a central symmetrical figure with the center of the circle as the symmetrical center. Theorem In the same circle or equal circle, the arcs of equal central angles are equal. The chord-to-chord distance of paired chords is equal to 1 15. It is deduced that if the distance between two central angles, two arcs, two chords or a set of chords of two chords is equal, the corresponding other quantity is equal to 1 16, and it is proved that the central angle of an arc is equal to half its central angle. In the same circle or in the same circle, the arc opposite to the equal circumferential angle is also equal to 1 18, and it is inferred that the circumferential angle opposite to the two semicircles (or diameters) is a right angle; The chord subtended by the 90 circumferential angle is 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 the inscribed quadrilateral of the diagonal complement and theorem circle of a right triangle 120. And any external angle is equal to its internal diagonal 12 1, ① straight line l intersects with ⊙O, D ⊙ ② straight line l is tangent to ⊙O, and D = R ⊙ 3 straight lines l and ⊙O and D ⊙/kloc-0. The property theorem of tangents The tangent of a circle is perpendicular to the tangent radius 124, the inference 1 a straight line passing through the center and perpendicular to the tangent must pass through the tangent point 125, the inference 2 a straight line passing through the tangent point and perpendicular to the tangent must pass through the center 126, and the tangent length theorem leads to two tangents of the circle from a point outside the circle. The length of their tangents is equal to the center of the circle, the straight line connecting this point bisects the included angle of the two tangents, the sum of the two opposite sides of the circumscribed quadrangle of the circle is equal to 128, and the tangent angle theorem is equal to the circumferential angle of the arc pair it clamps. Therefore, if the arcs clamped by the two tangent angles are equal, the two tangent angles are also equal to 130. The product of the length of two lines divided by the intersection is equal to 13 1. It is deduced that if the chord intersects the diameter vertically, then half of the chord is the median term of the ratio of the two line segments formed by its diameter, and the tangent theorem draws the tangent and secant of the circle from a point outside the circle. The tangent length is the term 133 of the ratio of the lengths of two lines from this point to the intersection of the secant and the circle. It is inferred that the product of the length of two lines from a point outside the circle to the intersection of each secant and the circle is equal to 134. If two circles are tangent, then the tangent point must be on the line 135. ① Two circles are circumscribed by D+R; ② Two circles are circumscribed by d=R+r; ③ Two circles intersect R-R-D-R+R (R-R); ④ Two circles are inscribed by D = R-R (R) theorem, and the intersection line of two circles bisects two vertically. Theorem divides the circle into n (n ≥ 3): (1)(2) The tangent of the circle passing through the point, and the polygon whose vertex is the intersection of adjacent tangents is the circumscribed regular n polygon of the circle 138. Theorem Any regular polygon has an outer polygon. These two circles are concentric circles 139, and each inner angle of a regular N-polygon is equal to (n-2) × 180/N 140, which proves the radius of the regular N-polygon. apome divides the regular N-polygon into 2n congruent right triangles14/kloc-. 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, k × (n-2) 180/n = 360 becomes (n-2)(k-2)=4 144, and the calculation formula of arc length is L = nR//kl. Outer common tangent length = d-(R+r) III. Formula classification Formula expression multiplication and factorization A2-B2 = (a+b) (a-b) A3+B3 = (a+b) (A2-AB+B2) A3-B3 = (A-B (A2+AB+B2)) Solution of a quadratic equation with one variable-B+√ (B2-4ac). Kloc-0/ * x 2 = c/A Note: The sum of the first n items in some series = N2 2+4+6+8+10+12+14+…+(2n) = n (n+/Kloc-0 ) (2n+1)/613+23+33 sinc = 2r Note: where r represents the radius of the circumscribed circle of the triangle, and cosine theorem b2=a2+c2-2accosB Note: Angle B is the angle between side A and side C, and there is an angle bisector in the compilation of auxiliary lines commonly used in junior high school geometry, which can be used as the bisector to both sides. You can also look at the picture in half, and there will be a relationship after symmetry. Angle bisector parallel lines, isosceles triangles add up. Angle bisector plus vertical line, try three lines. Perpendicular bisector is a line segment that usually connects the two ends of a straight line. It needs to be proved that the line segment is double-half, and extension and shortening can be tested. The two midpoints of a triangle are connected to form a midline. A triangle has a midline and the midline extends. A parallelogram appears and the center of symmetry bisects the point. Make a high line in the trapezoid and try to translate a waist. It is common to move diagonal lines in parallel and form triangles. The card is almost the same, parallel to the line segment, adding lines, which is a habit. In the proportional conversion of equal product formula, it is very important to find the line segment. Direct proof is more difficult, and equivalent substitution is less troublesome. Make a high line above the hypotenuse, which is larger than the middle term. Calculation of radius and chord length, the distance from the chord center to the intermediate station. If there are all lines on the circle, the radius of the center of the tangent point is connected. Pythagorean theorem is the most convenient for the calculation of tangent length. To prove that it is tangent, carefully distinguish the radius perpendicular. Is the diameter, in a semicircle, to connect the chords at right angles. An arc has a midpoint and a center, and the vertical diameter theorem should be remembered completely. There are two chords on the corner of the circle, and the diameters of the two ends of the chords are connected. Find tangent chord, same arc diagonal, etc. If you want to draw a circumscribed circle, draw a vertical line in the middle on both sides. Also make an inscribed circle, and the bisector of the inner corner is a dream circle. If you meet an intersecting circle, don't forget to make it into a string. Two circles tangent inside and outside pass through the common tangent of the tangent point. If you add a connector, the tangent point must be on the connector. Adding a circle to the equilateral angle makes it not so difficult to prove the problem. The auxiliary line is a dotted line, so be careful not to change it when drawing. If the graph is dispersed, rotate symmetrically to carry out the experiment. Basic drawing is very important and should be mastered skillfully.