3. Seva theorem

4. Inverse theorem of Seva theorem

5. Two inferences of the generalized Pythagorean theorem;

Inference: The sum of squares of the diagonal of a parallelogram is equal to the sum of squares of four sides.

6. Theorem of bisector of triangle inner angle and outer angle;

7. Ptolemy theorem

8. Triangle situation center theorem

9. Sine theorem

10. Cosine theorem

1 1. siemsen theorem

12. euler theorem

13. bass plus straight line 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.

8 Inverse Theorem The point where the distance between the two endpoints of a line segment is equal 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.

10 Theorem 1 Two graphs symmetric about a straight line are conformal.

1 1 Theorem 2 If two graphs are symmetrical about a straight line, then the symmetry axis is the perpendicular line connecting the corresponding points.

12 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.

13 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.

14 Pythagorean Theorem The sum of squares of two right angles A and B of a right triangle is equal to the square of the hypotenuse C, that is, A+B = C.

Inverse Theorem of Pythagorean Theorem 15 If the lengths of three sides of a triangle A, B and C are related to a+b=c, then the triangle is a right triangle.

Theorem 16 The sum of the inner angles of the quadrilateral is equal to 360, and the sum of the outer angles of the quadrilateral is equal to 360.

17 Theorem The sum of the internal angles of a polygon and an n-sided polygon is equal to (n-2) × 180.

18 infers that the sum of the external angles of any polygon is equal to 360.

19 parallelogram property theorem: the diagonals of parallelogram are equal, and the opposite sides of parallelogram are equal.

It is inferred that the parallel segments sandwiched between two parallel lines are equal.

2 1 parallelogram property theorem The diagonal of a parallelogram is equally divided.

22 parallelogram decision theorem 1 Two groups of parallelograms with equal diagonals are parallelograms.

23 parallelogram decision theorem 2 Two groups of parallelograms with equal opposite sides are parallelograms.

24 parallelogram decision theorem 3 quadrilaterals whose diagonals are bisected are parallelograms.

25 parallelogram decision theorem 4 A group of parallelograms with equal opposite sides are parallelograms.

26 Rectangular Decision Theorem 2 A parallelogram with equal diagonals is a rectangle.

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

28 Diamond Property Theorem 2 Diagonal lines of the diamond are perpendicular to each other, and each diagonal line bisects a set of diagonal lines.

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

Theorem of 30 1 is congruent on two centrosymmetric graphs.

3 1 Theorem 2 For two graphs that are symmetric about the center, the connecting lines of symmetric points both pass through the symmetric center and are equally divided by the symmetric center.

32 Inverse Theorem If a straight line connecting the corresponding points of two graphs passes through a point and is equally divided by the point, then the two graphs are symmetrical about the point.

33 isosceles trapezoid property theorem The two angles of isosceles trapezoid on the same base are equal.

The two diagonals of an isosceles trapezoid are equal.

35 isosceles trapezoid decision theorem A trapezoid with two equal angles on the same base is an isosceles trapezoid.

A trapezoid with equal diagonal lines is an isosceles trapezoid.

37 Theorem of Equal Segment 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.

38 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.

The median line theorem of 40 triangles is parallel to the third side and equal to half of it.

The midline theorem of 4 1 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÷2 S=L×h).

42 basic properties of the ratio If a:b=c:d, then ad=bc If ad=bc, then A: B = C: D.

43 parallel lines are divided into segments and the theorem of proportionality. Three parallel lines cut two straight lines, and the corresponding segments are proportional.

It is inferred that the line parallel to one side of the triangle cuts the other two sides (or the extension lines of both sides), and the corresponding line segments are proportional.

Theorem 45 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.

46 is a straight line parallel to one side of the triangle and intersecting with the other two sides. The three sides of the cut triangle are proportional to the three sides of the original triangle.

Theorem 47 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.

48 Judgment Theorem of Similar Triangle 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.

If the median line of one side of a triangle is equal to half of this side, then this triangle is a right triangle.

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

5 1 Decision Theorem 3 Three sides are proportional and two triangles are similar (SSS)

Theorem 52 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.

53 Property Theorem 1 similar triangles corresponds to the height ratio, the ratio corresponding to the median line and the ratio corresponding to the angular bisector are all equal to the similarity ratio.

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

55 Property Theorem 3 The ratio of similar triangles area 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.

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.

A circle is a set of points whose distance from a fixed point is equal to a fixed length.

The interior of a circle can be regarded as a collection of points whose center distance is less than the radius.

The outside of a circle can be regarded as a collection of points whose center distance is greater than the radius.

6 1 The distance to a 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 a point whose distance is equal to the two ends of a known line segment is the median vertical line of the line segment.

The locus of a point with equal distance on both sides of a known angle is the bisector of this angle.

The locus from 64 to the point with equal distance between two parallel lines is a straight line parallel to these two parallel lines and with equal distance.

Theorem 65 determines the straight line between three points that are not on the same straight line.

The vertical diameter theorem divides the chord perpendicular to the chord diameter into two parts, and divides the two arcs opposite the chord into two parts.

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

Inference 2 The arcs sandwiched between two parallel circular chords are equal.

A circle is a central symmetrical figure with the center of the circle as the symmetrical center.

Theorem 70 In the same circle or in the same circle, the isocentric angle has equal arc, chord and chord center distance.

7 1 Inference: In the same circle or equal circle, if one set of quantities in two central angles, two arcs, two chords or chord-center distances of two chords are equal, the corresponding other set of quantities are equal.

72 Inference: 1 Same arc or equal arc with the same circumferential angle; In the same circle or in the same circle, the arcs of equal circumferential angles are also equal.

73 Inference 2 The circumference angle (or diameter) of a semicircle is a right angle; A chord with a circumferential angle of 90 is a diameter.

Theorem 74 Diagonal lines of inscribed quadrangles of a circle are complementary, and any external angle is equal to its internal angle.

75① The line L intersects with ⊙O D￠r ② The line L is tangent to ⊙O d=r ③ The line L is separated from ⊙ o d ￠ r.

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.

The property theorem of tangent The tangent of a circle is perpendicular to the radius passing through the tangent point.

78 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 point and perpendicular to the tangent line must pass through the center of the circle.

The tangent length theorem leads to two tangents of a circle from a point outside the circle, and their tangent lengths are equal. The line between the center of the circle and this point bisects the included angle of the two tangents.

The sum of two opposite sides of the circumscribed quadrilateral of 8 1 circle is equal.

Chord tangent angle theorem Chord tangent angle is equal to the circumferential angle of the arc pair it clamps.

It is inferred that if the arc enclosed by two chord angles is equal, then the two chord angles are also equal.

Theorem of Intersecting Chords The length of two intersecting chords in a circle divided by the product of the intersection point is equal.

It can be inferred that if the chord intersects the diameter vertically, then half of the chord is the proportional average of the two line segments formed by its diameter.

The tangent theorem leads to the tangent and secant of a circle from a point outside the circle, and the tangent length is the middle term in the length ratio of the two lines at the intersection of this point and secant.

It is inferred that the product of two secant lines from a point outside the circle to the intersection of each secant line and the circle is equal.

If two circles are tangent, then the tangent point must be on the line.

89 ① Perimeter of two circles D+R ② Perimeter of two circles d=R+r ③ Intersection of two circles R-R-D+R (R+R).

④ two circles are inscribed with d = R-R(R￠R)⑤ two circles contain d￠R-R(R￠R).

Theorem 90 The intersection of two circles bisects the common chord of two circles vertically.

(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 9 1 Any regular polygon has a circumscribed circle and an inscribed circle, which are concentric circles.

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

Theorem 93 The radius and vertex of a regular N-polygon divide the regular N-polygon into 2n congruent right triangles.

The area sn = pnrn/2 p of 94 regular n-polygons represents the perimeter of the regular n-polygon.

95 The area of a regular triangle √ 3a/4a indicates the side length.

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.

98 arc length calculation formula: L = n ∏ R/ 180.

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

100 inner common tangent length = d-(R-r) outer common tangent length = d-(R+r)

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