Many people have been making auxiliary lines for many years, and they have not thought clearly about the purpose of making auxiliary lines. In fact, the purpose of auxiliary lines is to establish connections between known topics.

There are many ways to make auxiliary lines, and the specific problems should be analyzed in detail, but there are also his own routines. I found this for you from other places. It's very comprehensive.

(1) triangle:

① Isosceles δ: the bisector of the height or vertex angle that is often connected with the bottom edge (construct two congruent right angles δ, or conveniently use the property that the three lines of isosceles δ are one. As shown in figure 1

② There is a midpoint δ on the hypotenuse of the right angle: connecting the midline (constructing two isosceles δ's, or conveniently using the special properties of the midline on the hypotenuse of the right angle δ's). As shown in fig. 2)

③ Oblique δ has a midpoint or midline: connecting the midline (constructing two equal product δ with the same bottom and the same height. As shown in fig. 3); Or perpendicular to the center line from the left and right vertices (construct two congruent right triangles. As shown in fig. 4); Either connect the midline, or take a parallel line from a midpoint as the other side (construct two similar δ, similarity ratio 1: 2, or conveniently use the δ midline theorem. As shown in figs. 5 and 6); Or double the midline or midline (construct two congruent δ's or complete a parallelogram). As shown in figs. 7 and 8). Or extend the midline by 1/3 (construct two congruent δ's or complete them as parallelograms. As shown in fig. 9).

(4) Angular bisector: the vertical line on both sides of the angle (forming an congruent right angle δ) through an intersection point on it. As shown in figure 10) or parallel lines on one side or both sides (construct one or two isosceles δ or rhombus. As shown in figure 1 1).

⑤ Angle bisector: On one side of this angle, take a section equal to the other side from the vertex and make a related line (construct two congruent δ). As shown in figure 12 and 13)

⑥ The bisector intersects the vertical line: the vertical line is often extended (construct isosceles δ. As shown in figure 14).

(2) Trapezoidal:

(1) Extend the waist to a point (the structure is similar to δ. As shown in figure 15),

(2) Make a waist parallel line from one end of the sole (construct a delta and a parallelogram by the difference between two waists and the upper and lower soles). As shown in figure 16).

(3) Both ends of the sole are perpendicular to the sole (two right angles δ and a rectangle are constructed. As shown in figure 17).

(4) When there is a diagonal: a parallel line with one end of the small bottom as the other diagonal (a delta and a parallelogram are formed by the sum of the two diagonal lines and the upper and lower bottom). As shown in figure 18).

(5) Connect one end of the small sole with the midpoint of the other waist and intersect with the extension line of the big waist (construct an congruent triangle and a trapezoid triangle with equal product). As shown in figure 19).

⑥ A parallel line with the midpoint of one waist as the other waist (and trapezoid form a parallelogram with equal delta and equal product. As shown in fig. 20).

⑦ Make parallel lines of two waists at the midpoint of the small bottom (use the difference between two waists in a concentration and the upper and lower bottoms to construct δ and two parallelograms. As shown in figure 2 1).

(3) Circle:

(1) Chord: the radius connecting the two ends of the chord, the diameter perpendicular to the chord or the distance between the centers of the chord (constructing a right angle δ is convenient for solving problems by using vertical diameter theorem, pythagorean theorem and acute trigonometric function); Or make one end of the chord tangent to the relevant central angle and circumferential angle (it is convenient to use the tangent angle theorem. As shown in fig. 22).

(2) A chord or semi-chord with a diameter and a vertical diameter, connecting the ends of the chord and the diameter (to construct three similar right-angle δ, it is convenient to use the properties and projective theorem of right-angle δ. As shown in fig. 23).

(3) quadrilateral inscribed with a circle: diagonal (construct more equal circumferential angles. As shown in fig. 24); Or extend one side of a quadrilateral (construct an outer angle equal to the inner diagonal). As shown in fig. 25).

(4) There is a tangent outside the circle: the radius or diameter passing through the tangent point (constructing a vertical relationship); Or a chord with a tangent point and the related central angle and circumferential angle (it is more convenient to use the chord tangent angle theorem. As shown in fig. 26).

⑤ There are two intersecting tangents outside the circle: the radius connecting the tangents, the intersection point connecting the tangents and the center of the circle (forming an congruent right triangle); Or the secant that has been crossed and added (convenient to use the tangent secant theorem); Or connect two tangent points (construct an isosceles δ, three pairs of congruent right angles δ, and bisect a chord vertically by the connecting line between the intersection of tangent lines and the center of the circle, which is convenient to use isosceles δ, right angles δ, congruent δ and projective theorem. As shown in fig. 27).

⑥ Secant \ A tangent line intersecting chords or intersecting outside a circle: it connects the endpoints of different chords or the intersections of different secants on a circle (the structure is similar to δ, so it is convenient to use the proportional line segment and δ exterior angle theorem. As shown in figs. 28, 29 and 30).

⑦ Intersection of two circles: make a connecting line, a chord or even a connecting line from the center of the two circles to both ends of the chord (Structure 2).

The isosceles δ is complementary to the positive form, which is convenient to apply the theorem that the connecting line bisects the common chord vertically. As shown in figure 3 1).

⑧ External tangents of two circles: connecting lines, internal tangents, external tangents, connecting points and connecting radii (there are two chords and external tangents in the structure set).

A right angle δ, a right trapezoid with a radius of two circles, the radius and the length of the tangent line. As shown in fig. 32).

⑨ Two circles are inscribed: make an intersection line and a tangent line of the intersection line (it is convenient to use the vertical relationship between the intersection line and the tangent line of the intersection line. As shown in fig. 33).

⑩ Two circles are separated from each other: parallel lines connected with each other and a common tangent or an internal common tangent, and the centers of small circles are connected with each other as a common tangent (construct a right angle δ which is concentrated on the length of the connecting line, the length of the common tangent and the difference or sum of the radii of the two circles. As shown in figs. 34 and 35).

It is known that there is a middle line in the 1 diagram, and the double long middle lines connect the straight lines.

The rotating structure is conformal, and the angle of equal line segment can be replaced.

The midline can be obtained by connecting multiple midlines with the midpoint.

If the bisector of an angle is known, the two sides can be perpendicular.

It can also be folded along the line to present congruent graphics.

If you add a vertical line to the bisector, you can see an isosceles triangle.

The angular bisector and parallel lines change the angular position of the equal line segment.

The vertical line in the line segment is known, which connects two equal line segments.

Some people say that geometry is difficult, and it is difficult in auxiliary lines.

Auxiliary line, how to add it? Master theorems and concepts.

We must study hard and find out the rules by experience.

There is an angle bisector in the problem, which can be perpendicular to both sides.

The perpendicular bisector of a line segment can connect the two ends of a straight line.

The two midpoints of a triangle are connected to form a midline.

A triangle has a midline, and the extended midline is equal in length.

Proportional, completely similar, often parallel lines.

If all the lines are outside the circle, they are tangent to the center of the circle to connect them.

If two circles are inscribed inside and outside, they will be tangent through the tangent point.

When two circles intersect at two points, they are generally called chords.

This is a diameter, this is a semicircle, and I want to make a right angle to connect the lines.

Make an equal angle and add a circle to prove that the problem is not that difficult.

The auxiliary line is a dotted line, so be careful not to change it when drawing.

There is an angular bisector in the picture, which can be perpendicular to both sides.

You can also look at the picture in half, and there will be a relationship after symmetry.

Angle bisector parallel lines, isosceles triangles add up.

Angle bisector plus vertical line, try three lines.

Perpendicular bisector is a line segment that usually connects the two ends of a straight line.

It needs to be proved that the line segment is double-half, and extension and shortening can be tested.

The two midpoints of a triangle are connected to form a midline.

A triangle has a midline and the midline extends.

A parallelogram appears and the center of symmetry bisects the point.

Make a high line in the trapezoid and try to translate a waist.

It is common to move diagonal lines in parallel and form triangles.

The card is almost the same, parallel to the line segment, adding lines, which is a habit.

In the proportional conversion of equal product formula, it is very important to find the line segment.

Direct proof is more difficult, and equivalent substitution is less troublesome.

Make a high line above the hypotenuse, which is larger than the middle term.

Calculation of radius and chord length, the distance from the chord center to the intermediate station.

If there are all lines on the circle, the radius of the center of the tangent point is connected.

Pythagorean theorem is the most convenient for the calculation of tangent length.

To prove that it is tangent, carefully distinguish the radius perpendicular.

Is the diameter, in a semicircle, to connect the chords at right angles.

An arc has a midpoint and a center, and the vertical diameter theorem should be remembered completely.

There are two chords on the corner of the circle, and the diameters of the two ends of the chords are connected.

Find tangent chord, same arc diagonal, etc.

If you want to draw a circumscribed circle, draw a vertical line in the middle on both sides.

Also make a dream circle with inscribed circle and bisector of inner angle.

If you meet an intersecting circle, don't forget to make it into a string.

Two circles tangent inside and outside pass through the common tangent of the tangent point.

If you add a connector, the tangent point must be on the connector.

Adding a circle to the equilateral angle makes it not so difficult to prove the problem.

The auxiliary line is a dotted line, so be careful not to change it when drawing.

If the graph is dispersed, rotate symmetrically to carry out the experiment.

Basic drawing is very important and should be mastered skillfully.

You should pay more attention to solving problems and often sum up the methods clearly.

Don't blindly add lines, the method should be flexible.

No matter how difficult it is to choose the analysis and synthesis methods, it will be reduced.

Study hard and practice hard with an open mind, and your grades will soar.