Today, I sorted out the geometric theorems that must be memorized in the senior high school entrance examination. These basic theorems are the key to solve geometric problems. Be sure to keep it in mind, and you can see more at ordinary times ~
Points, lines and angles
Theorem of points: there is only one straight line passing through two points.
Point theorem: the shortest line segment between two points.
Angle theorem: vertex angles are equal.
Angle theorem: the complementary angles of the same angle or equal angle are equal.
Angle theorem: the complementary angles of the same angle or equal angle are equal.
Straight line theorem: in the same plane, only one straight line passes through a point and is perpendicular to the known straight line.
Straight line theorem: Of all the line segments connecting a point outside a straight line with a point on a straight line, the vertical line segment is the shortest.
Geometric parallelism
Parallelism theorem: after passing a point outside a straight line, there is one and only one straight line parallel to this straight line.
Inference: If both lines are parallel to the third line, then the two lines are also parallel to each other.
Prove the parallel theorem of two straight lines: the same angle is equal and the two straight lines are parallel; Internal dislocation angles are equal and two straight lines are parallel; The internal angles on the same side are complementary and the two straight lines are parallel.
Inference that two straight lines are parallel: two straight lines are parallel and the same angle is equal; The two straight lines are parallel and the internal dislocation angles are equal; These two lines are parallel and complementary.
The sides and angles 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.
Theorem of sum of interior angles of triangle: the sum of three interior angles of triangle is equal to 180.
Congruent triangles decision
Theorem: The sides and angles corresponding to congruent triangles are equal.
Edge Theorem (SAS): Two triangles have two sides, and their included angles are congruent.
Angle Theorem (ASA): Two triangles have two angles and their sides are congruent.
Inference (AAS): Two triangles with two angles and opposite sides of one angle are congruent.
Edge Theorem (SSS): Two triangles corresponding to three equilateral sides are congruent.
Hypotenuse and right-angled edge theorem (HL): Two right-angled triangles with hypotenuse and a right-angled edge are congruent.
internal bisector
Theorem 1: The distance from a point on the bisector of an angle to both sides of the angle is equal.
Theorem 2: In an angle, the points with equal distance to both sides of the angle are on the bisector of the angle.
The bisector of an angle is the set of all points with equal distance to both sides of the angle.
Isosceles triangle property
The property theorem of isosceles triangle: the two base angles of isosceles triangle are equal (that is, equilateral angles)
Inference 1: The bisector of the vertex of an isosceles triangle bisects the base and is perpendicular to the base.
Unfolding: The bisector of the top angle, the median line on the bottom edge and the height on the bottom edge of the isosceles triangle coincide.
Judgment theorem of isosceles triangle: If the two angles of a triangle are equal, then the opposite sides of the two angles are also equal (equal angles and equal sides).
Symmetry 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.
Inverse theorem: the equidistant points of two endpoints of a line segment are on the middle vertical line of this line segment.
The median vertical line of a line segment can be regarded as a set of all points with equal distance to both ends of the line segment.
Theorem 1: congruence of two graphs symmetric about a line.
Theorem 2: If two figures are symmetrical about a straight line, then the symmetry axis is the perpendicular line connecting the corresponding points.
Theorem 3: Two figures 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.
Right triangle theorem
Theorem: 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.
The center line of the hypotenuse of a right triangle is equal to half of the hypotenuse.
Pythagorean theorem: the sum of squares of two right angles A and B of a right triangle is equal to the square of hypotenuse C, that is, A? +b？ = c？
Inverse theorem of Pythagorean theorem: If three sides of a triangle are related to A, B, C? +b？ = c？ Then this triangle is a right triangle.
Theorem of sum of interior angles of polygons
Theorem: The sum of internal angles of quadrilateral is equal to 360; The sum of the external angles of the quadrilateral is equal to 360.
Theorem of the sum of internal angles of polygons: the sum of internal angles of n polygons is equal to (n-2) × 180.
Inference: The sum of the external angles of any polygon is equal to 360.
Parallelogram theorem
Parallelogram property theorem;
1. The diagonals of the parallelogram are equal.
2. The opposite sides of the parallelogram are equal.
3. The diagonal of parallelogram is equally divided.
Inference: The parallel segments sandwiched between two parallel lines are equal.
Parallelogram judgment theorem:
1. Two sets of quadrangles with equal diagonals are parallelograms.
2. Two groups of quadrilaterals with equal opposite sides are parallelograms.
3. Quadrilaterals whose diagonals are bisected with each other are parallelograms.
4. A set of quadrilaterals with parallel and equal opposite sides is a parallelogram.
Rectangular theorem
Theorem of Rectangular Properties 1: All four corners of a rectangle are right angles.
Theorem 2 of Rectangular Properties: The diagonals of rectangles are equal
Rectangular Decision Theorem 1: A quadrilateral with three right angles is a rectangle.
Rectangular Decision Theorem 2: Parallelograms with equal diagonals are rectangles.
Diamond theorem
Diamond property theorem 1: all four sides of a diamond are equal.
Theorem 2: Diagonal lines of rhombus are perpendicular to each other, and each diagonal line bisects a set of diagonal lines.
Diamond area = half of diagonal product, that is, S=(a×b)÷2.
Diamond Decision Theorem 1: A quadrilateral with four equilateral sides is a diamond.
Diamond Decision Theorem 2: Parallelograms whose diagonals are perpendicular to each other are diamonds.
Square theorem
Theorem of Square Properties 1: All four corners of a square are right angles and all four sides are equal.
Theorem 2: Two diagonal lines of a square are equal and bisected vertically, and each diagonal line bisects a set of diagonal lines.
Central symmetry theorem
Theorem 1: congruence of two graphs with central symmetry.
Theorem 2: For two graphs with symmetrical centers, the connecting lines of symmetrical points pass through the symmetrical centers and are equally divided by the symmetrical 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.
isosceles trapezoid
Theorem of isosceles trapezoid properties;
1. The two angles of the isosceles trapezoid with the same base are equal.
2. The two diagonals of the isosceles trapezoid are equal.
Judgement theorem of isosceles trapezoid;
1. A trapezoid with two equal angles on the same base is an isosceles trapezoid.
2. A trapezoid with equal diagonal lines is an isosceles trapezoid.
Theorem of bisection 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.
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.
pappus law
The midline theorem of triangle: the midline of triangle is parallel to the third side and equal to half of it.
Trapezoidal mean value theorem: the midline of the trapezoid is parallel to the two bases and is equal to half of the sum of the two bases L=(a+b)÷2, S = L× H.
Similar triangles theorem
Similar triangles's Theorem: 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.
Similar triangles's judgment theorem;
1. Two angles are equal and two triangles are similar.
2. The two sides are proportional and the included angle is equal, and the two triangles are similar.
Two right triangles divided by the height on the hypotenuse are similar to the original triangle.
Decision Theorem 3: Three sides are proportional and two triangles are similar.
Theorem of Similar Right Triangle: If the hypotenuse and right side of one right triangle are proportional to the hypotenuse and right side of another right triangle, then the two right triangles are similar.
Attribute theorem:
1. similar triangles corresponds to the height ratio, the ratio corresponding to the center line and the ratio corresponding to the angular bisector are all equal to the similarity ratio.
2. The ratio of similar triangles perimeter is equal to similarity ratio.
3. The ratio of similar triangles area is equal to the square of similarity ratio.
Trigonometric function theorem
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.
Theorem of circle
Theorem: A three-point crossing line can be made into one and only one circle.
Theorem: The diameter perpendicular to the chord bisects the chord and bisects the two arcs opposite the chord.
Inference 1: bisect the diameter of the chord (not the diameter) perpendicular to the chord and bisect the two arcs opposite the chord.
Inference 2: The perpendicular bisector of a chord passes through the center of the circle and bisects the two arcs opposite to the chord.
Inference 3: bisect the diameter of an arc opposite to the chord, bisect the chord vertically and bisect another arc opposite to the chord.
Theorem:
1. In the same circle or equal circle, the chords of equal arcs are equal, and the chord center distances of the chords are equal.
2. The straight line passing through the outer end of the radius of a circle and perpendicular to the radius is the tangent of the circle.
3. The tangent of the circle is perpendicular to the radius passing through the tangent point.
4. The bisectors of the three internal angles of a triangle intersect at one point, which is the heart of the triangle.
5. Two tangents drawn from a point outside the circle are equal in length, and the connecting line between the center of the circle and the point bisects the included angle of the two tangents.
6. The sum of two opposite sides of a circle's circumscribed quadrilateral is equal.
Proportional property theorem
Basic properties of proportion
If a: b = c: d, then ad = BC If ad=bc, then A: B = C: D.
Combined ratio attribute
If a/b=c/d, then (a b)/b = (c d)/d.
Equidistant property
If a/b=c/d=…=m/n(b+d+…+n≠0), then (a+c+…+m)/(b+d+…+n) = a/b.