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geometric proof

A summary of the knowledge points of geometric proofs, including geometric proofs, perpendicular bisectors of line segments and angle bisectors, and right triangles. The content is substantial and the logic is clear.

Edited at 2023-01-20 20:25:30

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geometric proof

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Outline

argument geometry

geometric proof

propositions and proofs

Deductive proof: starting from known concepts and conditions, based on confirmed facts and recognized logical rules, the process of deducing that a certain conclusion is correct

Proposition: a sentence that judges something

Judgment correct: true proposition

Judgment error: false proposition

Steps to prove a proposition is true (give counterexamples to false propositions)

Draw pictures and label letters (converting textual language into graphic language)

Based on the question and conclusion, combined with graphics, write what is known and verified

Analyze the graph and write down the proof process

Axioms: true propositions that people summarize from long-term practice

A true proposition that starts from axioms or other true propositions, is proved to be correct by reasoning methods, and further serves as a basis for judging the truth or falsehood of other propositions.

Proof example

Add auxiliary lines

Construct basic graphics and special graphics

Movement of graphics: translation, rotation, flipping

Correctly describe the content of adding auxiliary lines

Perpendicular bisectors of line segments and bisectors of angles

Converse propositions and converse theorems

Reciprocal propositions: Among two propositions, if the proposition of the first proposition is the conclusion of the second proposition, and the conclusion of the first proposition is the proposition of the second proposition,

One of the propositions is called the original proposition, and the other proposition is called its converse proposition.

Reciprocity theorem: If the inverse proposition of a store is proved to be two theorems of the theorem

One of them is called the converse theorem of the other

perpendicular bisector of line segment

From the congruence theorem: the distance from any point on the perpendicular bisector of a line segment to both endpoints of the line segment is equal.

Add auxiliary lines to construct special graphics. The inverse theorem is obtained by combining three isosceles lines: a point that is equidistant from the two endpoints of a line segment is on the perpendicular bisector of the line segment.

bisector of angle

From congruence: the distance from any point on the bisector of an angle to both sides of the angle is equal

The converse theorem follows from the theorem: A point inside an angle (including the vertex) and equidistant from both sides of the angle lies on the bisector of the angle.

trajectory

Definition: The set of all points that meet certain conditions

Three basic trajectory graphics

Draw trajectory graphics: Intersection method, determine the two trajectories to draw the graphics

right triangle

determination

Judgment of general triangles: S.A.S;A.S.A;A.A.S;S.S.S

Determination of congruence of right-angled triangles: the hypotenuse and a right-angled side are correspondingly equal (H.L)

Through the movement of graphics, mathematical ideas from general to specific

Properties (analogy thinking, analogy to the study of isosceles triangles)

side

According to the shortest property of the vertical line segment, the theorem is drawn: In a right triangle, the hypotenuse is greater than the right side.

special relationship

Through the study of irrational numbers, four right triangles form a square with an area of 2

The square of the hypotenuse is 2 and the square of the right angle is 1

Using three squares of different sizes and four right triangles, cut and mend, you can get

Pythagorean Theorem: The sum of the squares of the two right-angled sides of a right triangle is equal to the square of the hypotenuse

Through puzzle shapes, calculations, using mathematics to combine ideas

Converse of the Pythagorean theorem

Construct a right triangle and use equal substitution to find congruence

If the square of one side of a triangle is equal to the sum of the squares of the other two sides, then the triangle is a right triangle

horn

From the sum of the interior angles 180 degrees and the angle 90 of a right triangle

Theorem 1: Two acute angles of a right triangle are complementary to each other

Properties in special right triangles

The acute angles are 30 degrees and 60 degrees respectively.

Add the auxiliary line to the center line of the hypotenuse to get a special figure. Use Theorem 2 to get the isosceles triangle and the equilateral triangle.

Properties of equilateral triangles, equivalent substitutions

Corollary 1: In a right triangle, if there is an acute angle equal to 30 degrees, then the right-angled side it opposes is equal to half of the hypotenuse

A right-angled side is equal to half the hypotenuse

Add auxiliary lines to obtain an equilateral triangle from special figures and Theorem 2

An equilateral triangle is 60 degrees. According to Theorem 1, the opposite angle is 30 degrees.

Corollary 2: In a right triangle, if there is a right-angled side equal to half of the hypotenuse, then the angle subtended by this right-angled side is equal to 30 degrees.

special line segment

When the acute angles are both 45 degrees, the midline on the hypotenuse is equal to half of the hypotenuse

Add auxiliary lines to construct basic graphics, congruent

Theorem 2: The midline of the hypotenuse of a right triangle is equal to half of the hypotenuse

Symmetry (not studied)

distance formula between two points

Add auxiliary lines to draw a straight line perpendicular to the x-axis and y-axis and the Pythagorean Theorem of a right-angled triangle.

subtopic

Using the plane rectangular coordinate system and the Pythagorean theorem, the idea of combining numbers and shapes

From special to general