Theorem 7 1 1 is congruent with respect to two centrosymmetric graphs.
Theorem 2 About two graphs with central symmetry, the connecting lines of symmetric points both pass through the symmetric center and are equally divided by the symmetric 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.
The property theorem of isosceles trapezoid is that two angles of isosceles trapezoid on the same base are equal.
The two diagonals of an isosceles trapezoid are equal.
76 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.
Theorem of Equal Segment of Parallel Lines If a group of parallel lines have equal segments on a straight line, the segments on other straight lines also have equal segments.
79 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 8 1 triangle The median line of a triangle is parallel to the third side and equal to half of it.
The trapezoid midline theorem is parallel to the two bases and equal to half of the sum of the two bases L = (a+b) ÷ 2s = l× h.
Basic properties of ratio 83 (1) If a:b=c:d, then ad=bc. If ad=bc, then a: b = c: d.
84 (2) Combinatorial Properties If A/B = C/D, then (A B)/B = (C D)/D.
85 (3) Isometric property If A/B = C/D = … = M/N (where b+d+…+n≠0), then (A+C+…+M)/(B+D+…+N) = A/B.
86 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 88 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.
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 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 decision theorem 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.
Decision Theorem 2: Two sides are proportional and the included angle is equal, and two triangles are similar (SAS).
Decision Theorem 3 Three sides are proportional and two triangles are similar (SSS)
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 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.
97 Property Theorem 2 The ratio of similar triangles perimeter is equal to similarity ratio.
98 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.
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 center distance is less than the radius.
The outer circle of 103 circle can be regarded as a collection of points whose center distance 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.
106 and the locus of the point with the same distance between the two endpoints of the known line segment is the middle vertical line of the 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 determines a straight line from three points that are not on a straight line.
1 10 vertical diameter theorem divides the chord perpendicular to its diameter into two parts, and divides the two arcs opposite to the chord into two parts.
1 1 1 inference 1
(1) bisects the diameter (not the diameter) of the chord perpendicular to the chord and bisects 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, the isocentric angle has equal arc, chord and chord center distance.
1 15 It is inferred 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 is equal, the corresponding other set of quantities is also equal.
Theorem 1 16 The angle of an arc is equal to half its central angle.
1 17 Inference 1 The circumferential angles of the same arc or the same arc are equal; In the same circle or in the same circle, the arcs of equal circumferential angles are also equal.
1 18 Inference 2 The circumferential angle (or diameter) of a semicircle 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.
120 Theorem The diagonals of the inscribed quadrilateral of a circle are complementary, and any external angle is equal to its internal angle.
12 1
(1) the intersection of straight line l and ⊙O d < r
(2) the tangent of the straight line l, and ⊙ o d = r.
③ 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 The property theorem of tangent line The tangent line 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 passing through the tangent and perpendicular to the tangent must pass through the center of the circle.
The tangent length theorem 126 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.
127 The sum of two opposite sides of a circle's circumscribed quadrilateral is equal.
128 Chord Angle Theorem The chord 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.
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132 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.
133 It is inferred that two secant lines of the circle are drawn from a point outside the circle, and the product of the lengths of the two lines from that point to the intersection of each secant line and the circle is equal.
134 If two circles are tangent, then the tangent point must be on the line.
135
① the distance between two circles is d > r+r+r.
(2) circumscribed circle d d = r+r.
③ the intersection of two circles 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 two circles vertically.
Theorem 137 divides a circle into n (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.
140 Theorem Radius and apothem Divides a regular N-polygon into 2n congruent right triangles.
14 1 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.
142 inner common tangent length = d-(R-r) outer common tangent length = d-(R+r)
143 area formula: ①S positive δ =-× (side length) 2. -②S parallelogram = base × height. ③S diamond = base× height = -× (diagonal product) -④S circle =πR2. ⑤C circumference = 2πr⑤ arc length L =-.