Between 1 and two points, the segment is the shortest.

There is a straight line after two o'clock, and there is only one straight line.

3, the vertex angles are equal; The complementary angle (or complementary angle) of the same angle is equal; The complementary angles (or complementary angles) of equal angles are equal.

4. After passing a point outside or on a straight line, one and only one straight line is perpendicular to the known straight line.

5.( 1) passes through a point outside the known straight line, and one and only one straight line is parallel to the known straight line.

(2) If two straight lines are parallel to the third straight line, then the two straight lines are also parallel.

6. Determination of parallel lines:

(1) Isomorphism angles are equal, two straight lines are parallel (2) internal dislocation angles are equal, and two straight lines are parallel (3) with complementary internal angles.

7, the characteristics of parallel lines:

(1) Two straight lines are parallel and have the same angle; (2) Two straight lines are parallel with equal internal angles; (3) Two straight lines are parallel and their internal angles are complementary.

8. The nature of the bisector: the distance from the point on the bisector to both sides of the angle is equal.

Judgment of the bisector of an angle: the points at the same distance from both sides of an angle are on the bisector of this angle.

9. The nature of the vertical line in the line segment: the distance from the point on the vertical line in the line segment to the two endpoints of the line segment is equal.

Determination of the vertical line in a line segment: the point with the same distance to the two endpoints of a line segment is on the vertical line in this line segment.

Second, triangles and polygons.

10, related axioms and theorems in triangle;

The properties of (1) triangle external angles: ① One external angle of a triangle is equal to the sum of two internal angles that are not adjacent to it; (2) An outer angle of a triangle is larger than any inner angle that is not adjacent to it; ③ The sum of the external angles of the triangle is equal to 360.

(2) Theorem of the sum of triangle internal angles: the sum of triangle internal angles is equal to 180.

(3) The sum of any two sides of a triangle is greater than the third side.

(4) Triangle midline theorem: the midline of a triangle is parallel to the third side and equal to half of the third side.

1 1, related axioms and theorems in polygons;

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

(2) Theorem of the sum of external angles of polygons: the sum of external angles of any polygon is 360.

(3) Euler formula: number of vertices+number of faces-number of edges =2.

12, if the graph is symmetrical about a straight line, then the line segments connecting the corresponding points are vertically bisected by the symmetry axis.

13, related axioms and theorems in isosceles triangle;

(1) The two base angles of an isosceles triangle are equal.

(2) If the two angles of a triangle are equal, then the opposite sides of the two angles are also equal.

(3) The theorem of "three lines in one" of isosceles triangle: the bisector of the top angle, the median line on the bottom edge and the height on the bottom edge of isosceles triangle coincide, which is called "three lines in one" for short.

(4) All internal angles of an equilateral triangle are equal, and each internal angle is equal to 60.

(5) A triangle with three equilateral sides is called an equilateral triangle; An isosceles triangle with an angle equal to 600 is an equilateral triangle;

A triangle with three equal angles is an equilateral triangle.

14, right triangle related axioms and theorems:

(1) The two acute angles of a right triangle are complementary.

(2) Pythagorean theorem: The sum of squares of two right angles of a right triangle is equal to the square of the hypotenuse.

(3) Pythagorean Inverse Theorem: If the square of one side of a triangle is equal to the sum of the squares of the other two sides, then this triangle is a right triangle.

(4) The median line on the hypotenuse of a right triangle is equal to half of the hypotenuse.

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

Third, the special quadrilateral.

15, the properties of parallelogram:

(1) The opposite sides of the parallelogram are parallel and equal (2) The diagonals of the parallelogram are equal (3) The diagonals of the parallelogram are equally divided.

16, the determination of parallelogram:

(1) Two groups of parallelograms with parallel opposite sides are parallelograms.

(2) A set of quadrilaterals with parallel and equal opposite sides is a parallelogram.

(3) Two groups of quadrangles with equal opposite sides are parallelograms.

(4) Two groups of quadrangles with equal diagonal are parallelograms.

(5) The quadrilateral whose diagonals bisect each other is a parallelogram.

17, the distance between parallel lines is equal everywhere.

18, properties of rectangle:

(1) All four corners of the rectangle are right angles. (2) The diagonal lines of the rectangle are equal and equally divided.

19, rectangle judgment: (1) A parallelogram with one right angle is a rectangle (2) A quadrilateral with three right angles is a rectangle (3) A parallelogram with equal diagonals is a rectangle.

20, the nature of the diamond:

(1) All four sides of the diamond are equal. (2) The diagonal lines of the diamond are divided vertically, and each diagonal line divides a set of diagonal lines equally.

2 1, the judgment of rhombus: (1) A group of parallelograms with equal adjacent sides is rhombus; (2) A quadrilateral with four equal sides is a diamond; (3) The parallelogram with vertical diagonal is a diamond.

22, the nature of the square:

(1) All four corners of a square are right angles (2) All four sides of a square are equal.

(3) The two diagonals of a square are equal and equally divided vertically, and each diagonal bisects a set of diagonals.

23, the judgment of the square:

Diamonds with right angles are squares.

(2) A group of rectangles with equal adjacent sides is a square.

(3) Two rectangles with vertical diagonal lines are squares.

(4) Two diamonds with equal diagonals are squares.

Trapezoid: A set of quadrangles with parallel opposite sides and another set of quadrangles with non-parallel opposite sides are trapeziums.

24, isosceles trapezoid judgment:

(1) Two trapeziums with equal internal angles on the same base are isosceles trapeziums.

(2) Two trapeziums with equal diagonals are isosceles trapeziums.

25, the nature of the isosceles trapezoid:

(1) The two internal angles on the same base of an isosceles trapezoid are equal.

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

26. The center line of the trapezoid is parallel to the two bottom sides of the trapezoid and equal to half of the sum of the two bottom sides.

Fourth, the shape is the same.

27. Properties of similar polygons:

(1) The corresponding edges of similar polygons are proportional; (2) The angles corresponding to similar polygons are equal.

(3) The ratio of the perimeters of similar polygons is equal to the similarity ratio.

(4) The area ratio of similar polygons is equal to the square of the similarity ratio.

(5) The corresponding angles of similar triangles are equal, and the corresponding edges are proportional; Similar triangles's ratio corresponding to high and the ratio corresponding to the center line are all equal to the similar ratio; The ratio of similar triangles perimeter is equal to similarity ratio; The area ratio of similar triangles is equal to the square of the similarity ratio.

28, similar triangles's decision:

(1) If two angles of one triangle are equal to two angles of another triangle, then the two triangles are similar.

(2) If two sides of a triangle are proportional to two sides of another triangle, and the included angles are equal, then the two triangles are similar.

(3) If three sides of a triangle are proportional to three sides of another triangle, then the two triangles are similar.

29. The edges and angles corresponding to congruent polygons are equal respectively.

30, congruent triangles's decision:

(1) Two triangles are congruent (S.S.S) if their three sides correspond to each other.

(2) If two triangles have two sides and their included angles are equal, then the two triangles are congruent (S.A.S).

(3) Two triangles are congruent (A.S.A) if their two angles and their clamping edges are equal respectively.

(4) There are two angles, and the opposite sides of one angle respectively correspond to the coincidence of two equal triangles.

(5) Two right-angled triangles are congruent (H.L.) if their hypotenuse and one right-angled side are equal respectively.

Verb (abbreviation for verb) circle

3 1, (1) In the same circle or equal circle, if one group of two central angles, two arcs and two chords is equal, the corresponding other group is equal; (2) The circumferential angles of semicircles or diameters are all equal, equal to 90 (right angle);

(3) A chord with a circumferential angle of 90 is the diameter of a circle.

32. In the same circle, the circumferential angle of the same arc or equal arc is equal, which is equal to half of the central angle of the arc; Equal circumferential angles face equal arcs.

33. Three points that are not on the same straight line determine a circle.

34.( 1) The straight line passing through the outer end of the radius and perpendicular to the radius is the tangent of the circle (2) The tangent of the circle is perpendicular to the radius of the tangent point.

35. Two tangents of a circle can be drawn from a point outside the circle, and their tangents are equal in length. The connecting line between this point and the center of the circle bisects the included angle between the two tangents.

36. The inscribed quadrilateral of a circle is diagonally complementary, and the outer angle is equal to the inner diagonal.

37. Vertical diameter theorem and inference: the diameter perpendicular to the chord bisects the chord and bisects the opposite arc; The diameter (not the diameter) that bisects the chord is perpendicular to the chord and bisects the two arcs opposite the chord.

Sixth, transformation.

37. Axisymmetric: (1) Two figures symmetrical about a line are congruent; If two figures are symmetrical about a straight line, then the symmetry axis is the middle vertical line connecting the corresponding points; (2) Two figures are symmetrical about a straight line. If their corresponding line segments (or extension lines) intersect, the intersection must be on the axis of symmetry; (3) Two figures are symmetrical about a straight line. If their corresponding line segments (or extension lines) intersect, the intersection must be on the axis of symmetry; (4) 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.

38. Translation: (1) Translation does not change the shape and size of the graphics (that is, the two graphics are identical before and after translation); (2) The corresponding line segments are parallel and equal (or on the same straight line), and the corresponding angles are equal; (3) After translation, the line segments corresponding to two points are parallel (or on the same straight line) and equal.

39. Rotation: (1) Rotation does not change the shape and size of the graph (that is, the two graphs are the same before and after rotation) (2) The angles formed by the connecting line of any pair of corresponding points and the rotation center are equal (both are rotation angles) (3) The distances from the corresponding points to the rotation center are equal after rotation.

40. Central symmetry: (1) Two figures with central symmetry are congruent; (2) For two graphs with symmetrical centers, the straight line connecting the symmetrical points passes through the symmetrical centers; (3) If a straight line connecting the corresponding points of two graphs passes through a certain point and is equally divided by the point, then the two graphs are symmetrical about the point.

4 1, similarity: (1) If two graphs are not only similar, but also each group of straight lines corresponding to vertices pass through the same point, then such two graphs are called similarity graphs, and this point is called similarity center, and the similarity ratio at this time is also called similarity ratio; (2) The ratio of the distance between any pair of corresponding points on the similarity graph and the similarity center is equal to the similarity ratio.

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A collection of mathematical geometry theorems in junior high school

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