(1) When △EFP is translated to the left along the straight line L to the position shown in Figure 2, EP intersects AC at point Q, connecting AP and BQ. Guess and write the quantitative and positional relationship between BQ and AP. Please prove your guess.
E
F
P
A
l
C
B
Q
Figure 3
E
A
Q
B
F
C
P
l
Figure 2
(2) When Δ△ EFP is translated to the left along the straight line L to the position in Figure 3, the extension line of EP intersects the extension line of AC at point Q, connecting AP and BQ. Do you think the quantitative relationship and positional relationship between BQ and AP predicted in (1) are still valid? If yes, give proof; If not, please explain why.
A (English)
B
C (female)
P
l
Figure 1
Finally, think about the second level of the problem
2. We know that two triangles have two opposite angles, and one of them is not necessarily congruent, so under what circumstances will it be congruent?
(1) Reading and proving:
① Because these two triangles are right triangles, it is obvious that they are congruent (HL).
(2) Because these two triangles are acute triangles, they are congruent, which can be proved as follows:
It is known that △ABC and △ are acute triangles, AB=, BC=, ∠C =∞. It is proved that:
△ABC?△。 (Please complete the following certification process)
It is proved that BD⊥CA is in D,
Yes,. . . . . (Please continue to testify)
③ If these two triangles are obtuse triangles, can they also be proved to be congruent? (Please write a complete proof process like ②. )
(2) inductive statement: from (1), a correct conclusion can be drawn. Please write this conclusion.
Finally, think about the third level of the problem
As shown in figure a, the middle is an acute angle, and the point is a point on the ray, connecting,
Think of one side as a square on the right.
Answer the following questions:
If,,
① When the point is on a line segment (not coincident with the point), as shown in Figure B, what is the relationship between the line segments? Please provide a justification for the answer.
② When the point is the extension of the line segment, as shown in Figure C, does the conclusion in ① still hold? Why?
Tujia nationality
A
B
D
F
E
C
Figure b
A
B
D
E
C
F
tubing
A
B
D
C
E
Finally, think about the fourth level of the problem
Figure 1
A
F
B
C
E
D
(1) As shown in figure 1, place two right-angled triangular plates with a 45 angle, with point D on BC and the extension line connecting BE, AD and AD at point F. 。
Description: af ⊥ be.
A
B
D
C
E
Figure 2
F
(2) As shown in Figure 2, two right-angled triangular flat plates with an angle of 30 degrees are placed, with point D on BC, and the extension line connecting BE, AD and AD intersects with BE at point F. Are AF and BE vertical? And explain why.
Final Thinking Level 5
Two congruent equilateral triangles △ABC and △ACD are used to form a quadrilateral ABCD, and a 60-degree angle triangular ruler is superimposed on this quadrilateral, so that the vertex of the 60-degree angle of the triangular ruler coincides with point A, and both sides coincide with AB and AC respectively, and the triangular ruler is rotated counterclockwise around point A. ..
(1) What conclusions can BE drawn by observing or measuring the lengths of Be and CF when the two sides of the triangular ruler intersect with the two sides of the quadrilateral BC and CD at points E and F respectively (as shown in Figure A)? And explain the reasons;
(2) When the two sides of the triangular ruler intersect with the extension lines of the two sides BC and CD of the quadrilateral at points E and F respectively (as shown in Figure B), is the conclusion you got in (1) still valid? Briefly explain why.
Final thinking level 6
(1) as shown in figure11,in △ADE, AE=AD and ∠AED=∠ADE, ∠ EAD = 90, and EC and DB are equally divided into ∠AED and.
(2) Keep the position of △ ade unchanged, and rotate △ABC counterclockwise around point A to the position where AD and BE intersect at O in figure 1 1-2. Please judge the relationship between segment BE and CD and explain the reasons.
Figure 1 1- 1
Figure 1 1-2
O
Final Thinking Level 7
1. As shown in the figure, BD and CE are both the heights of △ABC. Cut BF from BD to make BF = AC, and take a point G from the extension line of CE to make CG = AB.
A
B
C
D
E
F
G
Try to discuss the relationship between AF and AG, and explain the reasons.
D
A
E
F
B
C
Figure (1 1)
2. As shown in figure (1 1), in equilateral, the points are on the sides respectively and intersect with the points.
(1) Verification:;
(2) Accuracy.
Final Thinking Level 8
1. Known: As shown in the figure, AB = CD, AD = BC, P is any point of AC, and the straight line passing through P intersects the extension lines of AD and CB at E and F respectively.
(1) Excuse me: ∠ E = ∠E=∠F? Explain your reasons;
(2) What conditions should be added to the conclusion that PE = PF, and explain your reasons.
A
B
C
D
E
F
P
2. Known: As shown in the figure, the line segment is known, and the two endpoints of the line segment are regarded as rays, so that the bisector of//intersects with the point, which is the midpoint of the line segment, and the intersection points are regarded as straight lines and rays, which intersect with the point respectively.
(1) description;
(2) Explain that the distance from a point to a straight line is equal.
Final thinking level 9
1. As shown in the figure, it is known that ∠AOB = 120, and OM divides ∠AOB equally. A vertex P of a regular triangle is placed on the ray OM, and both sides intersect with DA and OB at points C and D respectively.
(1) As shown in Figure ①, if the edge PC and OA are perpendicular, are the line segments PC and PD equal? Why?
(2) As shown in Figure ②, rotate a regular triangle around point P so that two sides intersect OA and OB at c ′ and d ′ respectively. Are the line segments PC' and PD' equal? Why?
、
2. It is known that the heights on both sides of △ABC and △DEF are △ABC and △DEF respectively. Try to discuss the relationship between them and explain the reasons.
The tenth level, the last thought.
(1) observation and discovery
Xiao Ming folds the triangular piece of paper along a straight line passing through point A, so that AC falls on the edge of AB, the crease is AD, and the piece of paper unfolds (as shown in Figure ①); Fold the triangle paper again, so that point A and point D coincide, and the crease is EF, which is obtained by flattening the paper (Figure ②). Xiao Ming thinks it is an isosceles triangle. Do you agree? Please explain the reason.
A
C
D
B
Figure ①
A
C
D
B
Figure ②
F
E
(2) Practice and application
Fold the rectangular piece of paper along a straight line passing through point B, so that point A falls on point F on the side of BC, and the crease is BE (as shown in Figure ③); Fold along the straight line passing through point E, so that point D falls on the point on BE, and the crease is EG (as shown in Figure ④); Flatten the paper again (Figure 5). Find the size in Figure ⑤.
E
Direct injury
C
F
B
A
Figure ③
E
D
C
A
B
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G
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Figure ④
Figure ⑤
Final thinking 1 1 level
As shown in figure 1, 2, 3, points e and d are the points on one extension line with point c as the vertex and the points on the other reverse extension line are in positive △ABC, regular quadrilateral ABCM and regular pentagonal ABCMN respectively (polygons with equal sides and angles are called regular polygons), and be = CD, and the DB extension line intersects AE at F.
(1) The number of times to find ∠AFB in figure 1;
(2) In Figure 2, the degree of ∠AFB is _ _ _ _ _ _, and in Figure 3, the degree of ∠AFB is _ _ _ _ _ _ _ _ _ _ _;
(3) According to the previous exploration, can this problem be extended to the general case of regular N polygons? If yes, write promotion questions and conclusions; If not, please explain why.
B
C
M
ordinary
D
E
F
three
A
A
B
C
D
E
1
F
B
M
A
F
2
E
D
C
Final thinking 12 level
It is known that ∠AOB=900, there is a point C on the bisector OM of ∠AOB, the right-angled vertex of the triangle coincides with C, and its two right-angled sides intersect OA and OB (or their reverse extension lines) at points D and E respectively.
When the triangle rotates around point C until CD is perpendicular to OA (as shown in figure 1), it is easy to prove that CD=CE.
When the triangle rotates around point C until CD is not perpendicular to OA, does the above conclusion still hold in both cases of Figure 2 and Figure 3? If yes, please give proof; If not, please write your guess and don't prove it.
Final thinking, 13.
As shown in the figure, △ABC and △ADC are congruent equilateral triangles. At the same time, point E and point F start from point B and point A respectively, and move to point A and point D in the directions of BA and AD respectively, with the same movement speed, connecting EC and FC.
(1) Does the size of ∠ECF change during the movement of points E and F? Please explain the reasons;
(2) During the movement of points E and F, does the area of the quadrilateral containing points A, E, C and F change? Please explain the reason.
A
E
B
C
D
F
(3) Connect EF, find all angles equal to ∠ACE in the graph, and explain the reasons.
(4) If points E and F continue to move on ray BA and ray AD, is the conclusion in (1) still valid? (Write the conclusion directly, without giving reasons)
Final thinking 14 level
Definition: A point with the same distance from one set of opposite sides of a convex quadrilateral and the same distance from another set of opposite sides is called a quasi-interior point of a convex quadrilateral. As shown in figure 1,,, then this point is the quasi-interior point of the quadrilateral.
Figure 3
Figure 2
Figure 4
F
E
D
C
B
A
P
G
H
J
I
Figure 1
B
J
I
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P
(1) As shown in Figure 2, the bisector of sum intersects at one point.
It is proved that a point is a quasi-interior point of a quadrilateral.
(2) Draw the quasi-interior points of the parallelogram in Figure 3 and the trapezoid in Figure 4 respectively.
There is no limit to drawing tools, and there is no writing method, but there must be necessary instructions. )
(3) To judge whether the following conclusions are correct, mark "√" for the right and "×" for the wrong.
① Any convex quadrilateral must have a quasi-interior point. ()
② Any convex quadrilateral must have only one quasi-interior point. ()
③ If it is a quasi-interior point of any convex quadrilateral, then or. ()
Final thinking level 15
(1) As shown in Figure 1, Figure 2 and Figure 3, in, they are regarded as edges respectively, and they are made into regular triangles, regular quadrangles and regular pentagons outward, and intersect at points.
Note: Polygons with equal sides and angles are called regular polygons.
① As shown in figure 1, verify that:
② Query: as shown in figure1; As shown in figure 2; As shown in figure 3.
(2) As shown in Figure 4, it is known that a group of adjacent sides are made into regular edges. It is the adjacent edge of a set of outward regular polygons. The extension of an edge intersects a point.
① conjecture: as shown in Figure 4, (expressed by inclusion);
Prove your guess according to Figure 4.