1

Addition of Common Auxiliary Lines in Triangle

Related to the angular bisector.

(1) can be perpendicular to both sides.

(2) Parallel lines can be used to construct isosceles triangles.

(3) Cut equal line segments on both sides of the corner to form an congruent triangle.

2. It is related to the length of the line segment.

(1) Truncation length: When it is proved that the sum or difference of two line segments is equal to the third line segment, a segment is often truncated on a longer line segment to make it equal to one of the line segments, and then the remainder is proved to be equal to the other line segment by congruence or similarity.

(2) Complementarity: When it is proved that the sum or difference between two line segments is equal to the third line segment, you can also extend a segment on the shorter line segment to make the extended part equal to another shorter line segment, and then prove that the extended line segment is equal to that longer line segment by congruence or similarity.

(3) Double long midline: If the midline of a triangle appears in the topic, the method is to double the midline and then connect the endpoints to get an congruent triangle.

(4) When the midpoint is met, consider the combination of the midline or three lines with equal sides.

3. Relating to an isosceles equilateral triangle

(1) Consider the fusion of three lines.

(2) Rotate a certain angle to construct all triangles. Isosceles generally rotate according to the degree of the vertex angle, while equilateral rotation is 60.

2

Addition of common auxiliary lines in quadrilateral

Special quadrangles mainly include parallelogram, rectangle, diamond, square and trapezoid. Auxiliary lines are often needed when solving some problems related to quadrilateral. Here are some ways to add auxiliary lines.

1. Practice of parallelogram related auxiliary lines

Parallelogram is one of the most common special quadrangles, and it has many useful properties. In order to make use of these properties, it is often necessary to add auxiliary lines to construct parallelograms.

(1) uses a set of parallelograms with equal sides.

(2) Two groups of opposite sides form a parallelogram.

(3) Construct a parallelogram by bisecting the diagonal.

2. Practice of having rectangular auxiliary lines

The calculation problem of (1) is generally solved by constructing a right triangle as an auxiliary line and using Pythagorean theorem.

(2) To prove or explore a problem, the diagonal lines connecting rectangles generally solve the problem by means of the equality of diagonal lines. There are few practices of auxiliary lines related to rectangles.

3. Diamond-related auxiliary line exercises

The auxiliary line related to the diamond is mainly the diagonal line connecting the diamond, and the problem is solved by the judgment theorem or property theorem of the diamond.

(1) is the height of the diamond.

(2) Diagonal lines connecting diamonds

4. Practice of auxiliary lines related to squares

A square is a perfect geometric figure, which is both axisymmetric and centrally symmetric. There are many questions about square. Sometimes it is necessary to make auxiliary lines to solve regular square problems, and the diagonal of the square is a common auxiliary line to solve square problems.

three

Addition of common auxiliary lines in a circle

1. When encountering a string (when solving the string problem)

Usually, the distance between the center of the chord is increased or made perpendicular to the radius (or diameter) of the chord or the radius of the end of the chord.

Function:

① Using the vertical diameter theorem

② Using the relationship between the central angle and its corresponding arc, chord and chord center distance.

(3) Form a right triangle with half of the chord, the chord center distance and the radius, and calculate the related quantity according to Pythagorean theorem.

2. When there is a diameter, the circumferential angle corresponding to the diameter is often added (drawn).

Function: Use the properties of rounded corners to get right-angled or right-angled triangles.

3. When encountering a 90-degree circumferential angle, two chords are usually connected to the other endpoint without a common point.

Function: The diameter can be obtained by using the properties of the circumferential angle.

4. When you meet a chord, you often connect the two endpoints of the chord at the center of the circle to form an isosceles triangle, or you can connect the two endpoints of the chord on the circumference.

Function: ① You can get an isosceles triangle.

(2) Equal fillets can be obtained according to the properties of fillets.

5. When there is a tangent, the radius of the tangent point (connecting the center of the circle and the tangent point) is often added.

Function: OA⊥AB can be obtained by using the property theorem of tangent, and right angle or right triangle can be obtained.

A point and a tangent point are usually added to the connecting circle.

Function: It can form the chord tangent angle, so as to apply the chord tangent angle theorem.

6. When encountering a tangent that proves that a straight line is a circle.

(1) If the common point between a straight line and a circle has not been determined, the center of the circle is often used as the vertical segment of the straight line.

Function: If OA=r, then L is tangent.

(2) If a straight line passes through a point on a circle, it connects this point with the center of the circle (i.e. radius).

Function: only need to prove OA⊥l, then L is tangent.

(3) If a point on or outside the circle is tangent to the circle.

7. When encountering two intersecting tangents (tangent length)

The tangent point is often connected with the center of the circle, the center of the circle is connected with a point outside the circle, and the two tangent points are connected.

Function: According to the tangent length and other properties, we can get

The equal relationship between (1) angle and line segment.

② Vertical relation

③ Congyu, similar triangles.

8. When encountering the inscribed circle of a triangle.

Connect the kernel to the vertex of each triangle, or make a vertical section of each side of the triangle through the kernel.

Function: Using the inner nature, you can get

① The straight line from the center to the three vertices of the triangle is the bisector of the triangle.

② The distance from the center to the three sides of the triangle is equal.

9. When encountering the circumscribed circle of a triangle, connect the circumscribed circle center and each vertex.

Function: The distance from the outer center to each vertex of the triangle is equal.

10. When two circles are separated (to solve the problem of external tangent and internal common tangent of two circles)

Often used as tangent point, connecting line, translating common tangent or translating radius of connecting line.

Function: ① Use the property of tangent; ② Using the knowledge of solving right triangle.

1 1. When two circles intersect, it is often used as a chord, connecting the lines between the two circles, connecting the intersection point and the center of the circle, etc.

Function: ① Use the properties of connecting lines to solve the knowledge about right triangle.

② Using the properties of quadrilateral inscribed in a circle

(3) Using the nature of the perimeters of two circles.

④ vertical diameter theorem

12. When two circles are tangent.

Often used as connecting lines and common tangents.

Function: ① Use the nature of the connection line.

② Tangent properties, etc.

13. When three circles are circumscribed.

Always make a line to connect every two circles.

Function: You can use the properties of the connection.

14. When the quadrangles are diagonally complementary or two triangles are at the same bottom, in the same direction at the bottom, and have equal "vertex angles".

Often add auxiliary circles.

Function: In order to make use of the characteristics of the circle.