Mathematical theorem
The relationship between the three sides of a triangle
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.
Sum of internal angles of triangle
The sum of the interior angles of a triangle is equal to 180.
It is inferred that the two acute angles of 1 right triangle are complementary.
Inference 2 An outer angle of a triangle is equal to the sum of two non-adjacent inner angles.
Inference 3: It's raining heavily in one outer corner of the triangle, and any inner corner that is not adjacent to it.
internal bisector
The point on the bisector of an angle is equal to the distance on both sides of the angle.
Geometric language:
∫OC is the bisector of ∝∠AOB (or ∝∠AOC =∠BOC).
PE⊥OA,PF⊥OB
Point p is on OC.
∴ PE = PF (property theorem of angular bisector)
The point where the decision theorem is equal to the distance between two sides of an angle is on the bisector of this angle.
Geometric language:
∵PE⊥OA,PF⊥OB
PE=PF
∴ Point P is on the bisector of ∞∠AOB (judgement theorem of bisector)
Properties of isosceles triangle
Property theorem of isosceles triangle; The two base angles of an isosceles triangle are equal.
Geometric language:
AB = AC
∴∠b =∞∠c (equilateral and equiangular)
Inference 1 The bisector of the vertex of the isosceles triangle bisects the base and is perpendicular to the base.
Geometric language:
( 1)∵AB=AC，BD=DC
∴∠ 1 =∠ 2, AD⊥BC (the bisector of the top angle of an isosceles triangle bisects the bottom vertically)
(2)∫AB = AC，∠ 1=∠2
∴AD⊥BC, BD = DC (bisector of the top angle of isosceles triangle bisects the bottom vertically)
(3)∵AB=AC,AD⊥BC
∴∠ 1 =∠ 2, BD = DC (bisector of the top angle of isosceles triangle bisects the bottom vertically)
Inference 2 All angles of an equilateral triangle are equal, and each angle is equal to 60.
Geometric language:
AB = AC = BC
∴∠a =∠b =∠c = 60 (all angles of an equilateral triangle are equal, and each angle is equal to 60).
Determination of isosceles triangle
Decision Theorem If the two angles of a triangle are equal, then the opposite sides of the two angles are also equal.
Geometric language:
∠∠B =∠C
∴ AB = AC (equilateral)
Inference 1 A triangle with three equal angles is an equilateral triangle.
Geometric language:
∠∠A =∠B =∠C
∴ AB = AC = BC (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.
Geometric language:
AB = AC, ∠ A = 60 (∠ B = 60 or ∠ C = 60).
∴ AB = AC = BC (an isosceles triangle with an angle equal to 60 is an equilateral triangle)
Inference 3 In a right-angled triangle, if an acute angle is equal to 30, then the right-angled side it faces is equal to half of the hypotenuse.
Geometric language:
∠∠C = 90 degrees, ∠B=30 degrees.
∴ BC = AB or AB = 2bc (in a right triangle, if an acute angle is equal to 30, then the right-angled side it faces is equal to half of the hypotenuse).
Perpendicular bisector of line segment
Theorem The distance between the point on the vertical line of a line segment and the two endpoints of this line segment is equal.
Geometric language:
∵MN⊥AB in C, AB = BC, (MN vertical AB)
Point p is any point in MN.
Pa = Pb (the perpendicular property of the line segment)
The inverse theorem and the point where the distance between the two ends of a line segment is equal are on the middle vertical line of this line segment.
Geometric language:
PA = PB
∴ Point P is on the perpendicular of line segment AB (judging from the perpendicular of line segment)
Axisymmetric and axisymmetric graphs
Theorem 1: Two graphs that are symmetric between a straight line are conformal.
Theorem 2 If two figures are symmetrical about a straight line, then the symmetry axis is the middle vertical line connecting the corresponding points.
Theorem 3 Two graphs 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.
pythagorean theorem
Pythagorean Theorem The sum of squares of two right-angled sides A and B of a right-angled triangle is equal to the square of the hypotenuse C, that is
a2 + b2 = c2
Inverse theorem of Pythagorean theorem
Inverse Theorem of Pythagorean Theorem If the lengths of three sides of a triangle A, B and C are related, then this triangle is a right triangle.
quadrilateral
Theorem The sum of the internal angles of any quadrilateral is equal to 360.
Sum of interior angles of polygons
The sum of the interior angles of Theorem Polygon and Theorem N Polygon is equal to (n-2) 180.
It is inferred that the sum of the external angles of any polygon is equal to 360.
Parallelogram and its properties
The property theorem of parallelogram 1 diagonal equality
Property Theorem 2 The opposite sides of parallelogram are equal
It is inferred that the parallel segments sandwiched between two parallel lines are equal.
Property Theorem 3 Diagonal lines of parallelograms are equally divided.
Geometric language:
∵ Quadrilateral ABCD is a parallelogram
∴AD‖BC, AB‖CD (the diagonals of parallelograms are equal)
∠ A =∠ C, ∠ B =∠ D (the opposite sides of a parallelogram are equal)
Ao = co, bo = do (the diagonal of parallelogram is equally divided)
Determination of parallelogram
Decision Theorem 1 Two groups of parallelograms with parallel opposite sides are parallelograms.
Geometric language:
In BC, BC
∴ quadrilateral ABCD is a parallelogram
(Two groups of parallelograms with parallel opposite sides are parallelograms)
Decision Theorem 2 Two sets of quadrangles with equal diagonals are parallelograms.
Geometric language:
∠∠A =∠C，∠B=∠D
∴ quadrilateral ABCD is a parallelogram
(Two sets of diagonally equal quadrilaterals are parallelograms)
Decision Theorem 3 Two sets of quadrilaterals with equal opposite sides are parallelograms.
Geometric language:
AD = BC，AB=CD
∴ quadrilateral ABCD is a parallelogram
(Two sets of quadrilaterals with equal opposite sides are parallelograms)
Decision Theorem 4 The quadrilateral whose diagonal bisects each other is a parallelogram.
Geometric language:
AO = CO，BO=DO
∴ quadrilateral ABCD is a parallelogram
(Quadrilaterals whose diagonals bisect each other are parallelograms)
Decision Theorem 5 A set of parallelograms whose opposite sides are parallel and equal is a parallelogram.
Geometric language:
∫AD‖BC，AD=BC
∴ quadrilateral ABCD is a parallelogram
(A set of quadrilaterals with parallel and equal opposite sides is a parallelogram)
rectangle
Property Theorem 1 All four corners of a rectangle are right angles.
Property Theorem 2 Diagonal lines of rectangles are equal
Geometric language:
∵ quadrilateral ABCD is a rectangle
∴ AC = BD (diagonal lines of rectangles are equal)
∠ ∠A =∠B =∠C =∠D = 90° (all four corners of a rectangle are right angles).
It is inferred that the median line on the hypotenuse of a right triangle is equal to half of the hypotenuse.
Geometric language:
∫△ABC is a right triangle, and ao = oc.
∴ bo = AC (the median line on the hypotenuse of a right triangle is equal to half of the hypotenuse)
Decision Theorem 1 A quadrilateral with three right angles is a rectangle.
Geometric language:
∠∠A =∠B =∠C = 90
The quadrilateral ABCD is a rectangle (a quadrilateral with three right angles is a rectangle)
Decision Theorem 2 A parallelogram with equal diagonals is a rectangle.
Geometric language:
AC = BD
∴ Quadrilateral ABCD is a rectangle (parallelogram with equal diagonal lines is a rectangle)
diamond
The four sides of the property theorem 1 diamond are equal.
Property Theorem 2 Diagonal lines of rhombus are perpendicular to each other, and each diagonal line bisects a set of diagonal lines.
Geometric language:
∫ The quadrilateral ABCD is a diamond.
∴ AB = BC = CD = AD (all four sides of the diamond are equal)
AC⊥BD, AC shares ∠DAB and ∠DCB, BD shares ∠ABC and ∠ADC.
(Diagonal lines of the diamond are perpendicular to each other, and each diagonal line bisects a set of diagonal lines)
Decision Theorem 1 A quadrilateral with four equilateral sides is a diamond.
Geometric language:
AB = BC = CD = AD
∴ quadrilateral ABCD is a diamond (a quadrilateral with four equilateral sides is a diamond)
Decision Theorem 2 Parallelograms whose diagonals are perpendicular to each other are diamonds.
Geometric language:
∵AC⊥BD,AO=CO,BO=DO
∴ Quadrilateral ABCD is a diamond (parallelograms with diagonal lines perpendicular to each other are a diamond)
square
Property Theorem 1 All four corners of a square are right angles and all four sides are equal.
Property Theorem 2 The two diagonals of a square are equal and bisected vertically, and each diagonal bisects a set of diagonals.
Centrally symmetric graphs and centrosymmetric graphs
Theorem 1 Two graphs symmetric about the center are conformal.
Theorem 2 For two graphs with symmetric centers, the connecting lines of symmetric points pass through the symmetric centers and are equally divided by the symmetric centers.
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.
trapeziform
The property theorem of isosceles trapezoid The two angles of isosceles trapezoid on the same base are equal.
Geometric language:
∵ quadrilateral ABCD is an isosceles trapezoid.
∴∠ A =∠ B, ∠ C =∠ D (two angles of an isosceles trapezoid on the same base are equal)
Decision theorem of isosceles trapezoid A trapezoid with two equal angles on the same base is an isosceles trapezoid.
Geometric language:
∠∠A =∠B，∠C=∠D
The quadrilateral ABCD is an isosceles trapezoid (a trapezoid with two equal angles on the same base is an isosceles trapezoid).
Triangular and trapezoidal neutral lines
The midline theorem of a triangle The midline of a triangle is parallel to the third side and equal to half of it.
Geometric language:
EF is the center line of the triangle.
∴ ef = ab (triangle median theorem)
The trapezoid midline theorem is parallel to the two bases and is equal to half the sum of the two bases.
Geometric language:
∫EF is the center line of the trapezoid.
∴ ef = (AB+CD) (Trapezoidal midline theorem)
Proportional line segment
1, the basic properties of proportion
If a ∶ b = c ∶ d, then ad = BC.
2, the proportion of nature
3. Isometric property
Proportional theorem of parallel line segment
Proportional theorem of dividing parallel lines into line segments Three parallel lines cut two straight lines, and the corresponding line segments are proportional.
Geometric language:
l‖p‖a
(Three parallel lines cut two straight lines, and the corresponding line segments are proportional. )
It is inferred that a straight line parallel to one side of a triangle cuts the other two sides (or extension lines on both sides), and the corresponding line segments are proportional.
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 of the triangle.
The diameter perpendicular to the chord.
The vertical diameter theorem bisects the chord perpendicular to its diameter and bisects the two arcs opposite the chord.
Geometric language:
∵OC⊥AB, OC passes through the center of the circle.
(vertical diameter theorem)
Inference 1
(1) bisects the diameter of the chord (not the diameter) perpendicular to the chord and bisects the two arcs opposite the chord.
Geometric language:
∵OC⊥AB, AC = BC, AB is not the diameter.
(bisecting the diameter of the chord (not the diameter) is perpendicular to the chord and bisecting 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.
Geometric language:
AC = BC, OC passes through the center of the circle.
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.
Geometric language:
(bisect the diameter of the arc opposite to the chord, bisect the chord vertically, and bisect the other arc opposite to the chord)
Inference 2 The arcs sandwiched by the two bisectors of a circle are equal.
Geometric language: ∫AB‖CD
The relationship between central angle, arc, chord and chord center distance
Theorem In the same circle or in the same circle, the isocentric angle has equal arc, chord and chord center distance.
It is deduced 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 are equal, the corresponding other set of quantities are equal respectively.
circumferential angle
Theorem The angle of an arc is equal to half of its central angle.
Inference 1 is equal to the circumferential angle of the same arc or equal arc; In the same circle or in the same circle, the arcs of equal circumferential angles are also equal.
Inference 2: the circumference angle (or diameter) of a semicircle is a right angle; A chord with a circumferential angle of 90 is a right angle.
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.
The inscribed quadrilateral of a circle
Theorem The diagonals of the inscribed quadrilateral of a circle are complementary, and any external angle is equal to its internal angle.
Geometric language:
∵ quadrilateral ABCD is an inscribed quadrilateral of⊙ O.
∴∠A+∠C= 180，∠B+∠ADB= 180，∠B=∠ADE
Determination and properties of tangent line
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.
Geometric language: ∵l ⊥OA, with point A on ⊙ O.
∴ The straight line L is tangent to⊙ O (tangent judgment theorem)
The property theorem of tangent The tangent of a circle is perpendicular to the radius passing through the tangent point.
Geometric language: ∵OA is the radius of ⊙O, and the straight line L cuts ⊙O at point A.
∴l ⊥OA (tangent property theorem)
Inference 1 The diameter 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.
Tangent length theorem
The theorem leads to two tangents of a circle from a point outside the circle, and their tangents are equal in length. The line between the center of the circle and this point bisects the included angle of the two tangents.
Geometric language: chords PB and PD cut ⊙O at points A and C.
∴PA=PC, ∠APO=∠CPO (tangent length theorem)
Angle of chord
Chord tangent angle theorem The chord tangent angle is equal to the circumferential angle of the arc pair it clamps.
Geometric language: ∵∠BCN is sandwich, ∠A is right.
∴∠BCN=∠A
It is inferred that if the arc enclosed by two chord angles is equal, then the two chord angles are also equal.
Geometric language: ∫∠BCN sandwiched, ∠ACM is right, =
∴∠BCN=∠ACM
Proportional line segment related to circle
Theorem of intersecting chords: the product of the length of two intersecting chords divided by the focus in a circle is equal.
Geometric language: chord AB and CD intersect at point p.
∴ pa Pb = PC PD (symphony theorem)
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.
Geometric language: ∵AB is the diameter, and CD⊥AB is at point P.
∴ pc2 = pa Pb (derivation of intersecting chord theorem)
The tangent theorem leads to the tangent and secant of a circle from a point outside the circle, and the tangent length is the median term of the ratio of the lengths of the two lines from this point to the secant and the focus of the circle.
Geometric language: ∵PT cuts ⊙ O at t point, and PBA is the secant of ⊙ o.
∴ pt2 = pa Pb (tangent theorem)
Inferred from the point outside the circle of two secant lines, the length of each secant line is equal to the product of the focus of the circle.
Geometric language: ∵PBA and PDC are secant of ∵ O.
∴ pt2 = pa Pb (derivation of the tangent line theorem
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