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Advanced Geometry

Excavate the essence of geometry and think about the rules, analyze the relationship between conclusions and conditions, first look at the distribution of conditions and the positional relationship with the conclusions, and secondly, look at whether there is a bridge between the conditions and the conclusions; all symmetrical elements are provably equal. , all parts are provably congruent, and congruence will definitely appear in symmetry. For optional questions, if all the available elements of symmetry are equal, the remaining corners can be directly considered to be equal (symmetry theorem).

Edited at 2022-01-23 09:29:06

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Advanced Geometry

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Outline

Junior high school geometry

Models and knowledge

Translation Rotation

One line and three equal angles

Basic conditions

isosceles triangle

Three equal angles on the same line

basic method

This type of problem often appears in the coordinate system, which is marked by an isosceles right triangle. When no ready-made model appears, draw vertical lines through the straight line at the vertex of the right triangle to the x and y axes.

rotate

diagonally complementary

Basic conditions (know two and push one)

angle bisector

diagonally complementary

A set of limbs are equal

basic method

do double dip

hand in hand

Basic conditions

Isosceles triangles with similar vertices

Basic conclusion

congruent

The angle between corresponding sides is equal to the angle of rotation

angular bisector

How to quickly find congruence

Angles with common vertices minus the overlapping part are congruent and the corresponding angles

Half-width model

Basic conditions (all are indispensable)

half angle relationship

Adjacent sides are equal

diagonally complementary

basic method

Cut an equilateral section outside the adjacent edge to construct congruence

parallel

angle bisector

Basic conditions (know two and infer four, but there must be parallelism)

parallel top and bottom

midpoint

There are two sets of angle bisectors

Bisectors of two angles are perpendicular

basic method

The essence is parallel, angle bisector, and isosceles. The method is to judge which one to push which one, and just extend the known angle bisector.

Parallel, isosceles and angle bisectors know two and push one.

Both exterior and interior angles

symmetry

The general drinks his horse

symmetry

Two moves are certain

Find the shortest length of two lines

Draw a symmetrical point of the fixed point, and then draw a vertical line

Find the shortest three lines

Make two symmetry points and connect them

Two must move

and minimum

Through a symmetrical point, it is connected to another point, and the straight line intersecting at one point is the moving point.

Biggest difference

According to the relationship between the three sides of the triangle, connect the two fixed points and extend the intersecting straight line to a point to become the moving point. If they are on different sides, they are converted to the same side.

Biggest difference

According to the center perpendicular, connect two points to draw the center perpendicular. The intersecting line at one point is the moving point.

translation symmetry

Single bridge repair problem

Translate the bridge length and convert it into a simple two-to-one movement on opposite sides (construct a parallelogram for translation)

Multiple

If the length of the bridge is moved out towards that straight line, then the straight line is equivalent to disappearing.

symmetry rotation

By holding hands together, congruence appears. Generally, the triangle containing the shorter side is rotated, and the points that are not on the straight line are transformed into a straight line, which is transformed into two definite and one moving. If the problem is still not solved, look at where the passive point is and move in a straight line

folded model

Find the angle

bisector of a double angle

The included angle is 90°

shoulder model

The sum of two adjacent supplementary angles is equal to the sum of the two vertex angles of the triangle

Axisymmetric + equilateral model

When there is a side that is symmetrical about a straight line in the question, and there is another side that is equal to this side (usually in a square or an isosceles triangle), pay attention to the equilateral side of the symmetrical side and the original side to form an isosceles triangle.

Find the edge

Special triangles can lead corners and edges

Three vertical models, congruent occurrences

Pythagorean theorem equation model

When a part of a fixed length a is folded and formed with another part to form a right triangle, the length of this part can be set to

Thinking rules

basic process

Big picture analysis

Analyze the relationship between the conclusion and the conditions. First, look at the distribution of the conditions and their positional relationship with the conclusion. Second, look at whether there is a bridge between the conditions and the conclusion.

Detail association

Fully connect the conditions with the rules of thinking, draw out all conclusions that can be drawn, and combine it with a dynamic perspective

Dynamic vision

Symmetry (appearance of corners, etc.)

Angle bisector (center perpendicular)

Theorem of angle bisector (perpendicular) and its converse theorem (making a double perpendicular) (connecting points and endpoints)

It is a symmetrical figure

Completion of symmetrical figures

All elements that are symmetrical are provable to be equal, and all parts are provable to be congruent. Symmetry must be congruent. For optional questions, if all the available elements of symmetry are equal, the remaining corners can be directly considered to be equal (symmetry theorem)

When you see a 45° angle, you think of an isosceles right triangle. When you see a 30° angle, you think of a 30° right triangle and their side relationships.

Rotate (edges appear, etc.)

Double length center line

See the midpoint, double the midline

Find the midpoint and draw parallel lines

Two congruent triangles have equal angles on their three sides. Typical models include tricongruent angles, hand in hand, and projective

Model

diagonal model

hand in hand

Half-width model

Pan

Parallel, isosceles and angle bisectors know two and push one.

Lead angle

When you need to find the relationship between angles at different positions, you can make parallel auxiliary lines to move the angles at different positions to the same vertex or triangle.

Leading edge

If the line segment is not in the triangle formed by the given length, or the distance is far away and it is difficult to find a relationship, then use the distance between parallel lines to be equal everywhere and translate it into the triangle formed by the given length.

symmetry rotation

Cut off the long and make up for the shortcomings

rotate

Two minor congruents containing shorter sides, existing sides, etc.

symmetry

There exists a complete set of angular-horizontal symmetries

Skill

Prove that the sides are congruent (prove that the sides are equal and find the length of the sides)

Check first to see if it is ready-made

When proving congruence, if a condition is missing. Check whether there is a missing corner or a missing side, list the possible situations, and then exclude those with circular logic, those that have nothing to do with the known, and those that do not satisfy the congruence determination theorem. Try the remaining ones one by one. You can put the missing corner or side in Prove from another set of congruent triangles

No ready-made

Consider constructing a set of congruent triangles containing this pair of sides/angles (first find out which complete triangle the existing equilateral sides are located in). The construction method is based on the congruence decision theorem. For example, if we already have a set of equal sides and a set of equal angles, we can construct another set of equal sides and equal angles to obtain a congruent triangle. If possible, construct the sides first.

Problem solving skills

When faced with several possible conclusions, you can guess which conclusion is correct by measuring, taking special values, taking limits, finding special positions (dynamic rotation changes), etc., and then start the next step of the proof.

When doing the second question or the second half of the problem-solving steps, you can contact the conclusion of the previous question and use it directly as a condition or imitate the previous conclusion and the method used to simplify the new problem.

knowledge thinking rules

If two figures have a common part, and the parts of the two figures except the common part are equal, then the two figures are equal.

If there are complex angle relationships, you can set parameters, set small angles, and find side relationships.

When the desired edge relationship has a coefficient and it is difficult to find the relationship, the coefficient can be proposed

In an uncertain graph, when it is necessary to classify and discuss, you can first determine the determined parts, and then discuss the possibilities on this basis.

Pythagorean theorem

Find the length

Similarity and congruence

Pythagorean theorem

Trigonometric functions

Formula for finding height (right triangle)

The product of two right angle sides divided by the base

When finding the length of a line segment, and the line segment is not in a right triangle, you can try to make a perpendicular line and include it in a right triangle.