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It is known that △ ABC and △ ade are equilateral triangles, and points a and E are on the same side of BC (1) As shown in Figure 1, if point D is on BC, write the quantitative relationship among line segments AC, CD and CE, and prove it; (2) as shown in Figure 2, if point D is on the extension line of BC, and other conditions remain unchanged, directly write the quantitative relationship among AC, CD and CE
(1) CD + CE = AC. the reasons are as follows: ∵ ABC is equilateral triangle, ∵ AB = AC = BC, ∵ BAC = 60 °, ∵ ade is equilateral triangle, ∵ ad = AE, ∵ DAE = 60 °, ∵ BAC - ∵ DAC = ∵ Dae - ? DAC, i.e. ∵ bad = ∵ CAE. In ∵ bad and ∵ CAE, ab = ad ? bad = caead = AE, ∵ ab ? bad = caead = AE
As shown in the figure, in equilateral △ ABC, point D is the midpoint of BC side, and ad is taken as the edge to make equilateral △ ade. (1) find the degree of ∠ CAE; (2) take the midpoint F of AB side, connect CF and CE, and try to prove that quadrilateral afce is rectangular
(1) ∵△ ABC is an equilateral triangle, and D is the midpoint of BC, which is bisected by ∵ Da, that is, ∵△ DAE is an equilateral triangle, and ∵ △ DAE is an equilateral triangle, with ∵ DAE = 60 °, and ∵ CAE = ∵ Dae - ∵ CAD = 30 °; (2) it is proved that ∵ △ BAC is an equilateral triangle, f is the midpoint of AB, CF ⊥ AB, BFC = 90
It is known that △ ABC is an equilateral triangle, the straight line a ‖ AB, D is a point on the straight line BC, one side De of ∠ ade intersects the straight line a at the point E, ∠ ade = 60 ° if D is on BC, the proof is: CD + CE = ca
It is proved that if cm = CD, ∵ ABC is equilateral triangle, ∵ ACB = 60 °, ∵ CDM is equilateral triangle, ∵ MD = CD = cm, ∵ CMD = ∵ CDM = 60 °, ∵ amd = 120 °, ∵ ade = 60 °, ∵ ade = ∵ MDC, ∵ ADM = ∵ EDC, ∥ straight line a ∥ AB, ∵ ace = ∥ BAC = 60 °
Triangle ABC, D is the midpoint of bottom BC, DF bisector angle ADB, de bisector angle ade, BF + CE is greater than EF
Because CE + cm is greater than em, BF + CE is greater than em. because of the bisectors of the two angles, so ∠ FDE = 1 / 2bdc = 90 ° because DF = DM (from the congruence above), so De is the middle perpendicular of MF
RELATED INFORMATIONS
- 1. As shown in the figure, D in △ ABC is any point on edge BC, de and DF are bisectors of △ ADB and △ ADC respectively, connecting EF. Try to judge the shape of △ def and explain the reason
- 2. In the triangle ABC, the angle BAC = 90 ° AB = AC, take a point D in the triangle to make the angle abd = 30 ° and BD = Ba, AE perpendicular to BD and e to find the bisector angle EAC of AD
- 3. As shown in the figure, in the triangle ABC, the angle ABC is 90 degrees and the angle c is 30 degrees. Rotate the triangle ABC counterclockwise around point a to the position of triangle ab'c ', and point B falls on AC The value of tancdb 'is?
- 4. In RT △ ABC, ∠ C = 90 degree, ∠ a = 30 degree, then BC: AC: ab = what
- 5. Δ ABC ≌ Δ CDA, AB and CD, BC and DA are corresponding edges. Write other corresponding edges and corresponding angles
- 6. ABC + CDA = ABCD ask what are ABCD In ABC, a is a hundred, B is a ten, C is a single, in CDA, C is a hundred, D is a ten, a is a single, in ABCD, a is a thousand, B is a hundred, C is a ten, D is a single
- 7. As shown in the figure, a, B and C are three points that are not on the same straight line, AA ′‖ BB ′‖ CC ′, and AA ′ = BB ′ = CC ′. Prove: plane ABC ‖ plane a ′ B ′ C ′
- 8. O is the outer center of the triangle ABC. If vector Ao multiplies vector AB = vector Bo multiplies vector BC, the shape of the triangle ABC can be helped by God
- 9. In △ ABC, the degrees of a and B are as follows, and () A. ∠A=50°，∠B=70°B. ∠A=70°，∠B=40°C. ∠A=30°，∠B=90°D. ∠A=80°，∠B=60°
- 10. As shown in the figure, it is known that P is a point on the side BC of △ ABC, and PC = 2PB. If ∠ ABC = 45 ° and ∠ APC = 60 °, find the size of ∠ ACB
- 11. In equilateral triangle ABC, is de / / BC. Angle ade an equilateral triangle?
- 12. Triangle ABC is an equilateral triangle, PA is perpendicular to plane ABC, and D is the midpoint of BC
- 13. Take any two numbers P and Q (P ≠ q) from the four numbers 2, 3, 4 and 5 to form the functions y = px-2 and y = x + Q, and make the image of these two functions If the intersection point is on the right side of the straight line x = 3, how many pairs of ordinal numbers are there in total?
- 14. Point a and B are two points on O, ab = 10, point P is a moving point on O (P does not coincide with ab) to connect AP, Pb passes through point o as OE ⊥ AP, of ⊥ Pb, and point EF is (1) Find the distance from the long (2) point O of the line EF to AB, and find the radius of O
- 15. If vector AP = 1 / 3 vector Pb and vector AB = Λ vector BP, then the value of Λ is?
- 16. Proof: as shown in Figure 9, in the triangle ABC, ab = AC,
- 17. As shown in the figure, in △ ABC, ∠ a = 36 °, ABC = 40 °, be bisection ∠ ABC, ∠ e = 18 °, CE bisection ∠ ACD? Why?
- 18. In △ ABC, ∠ ACB = 90 ° ch ⊥ AB in H, △ ACD and △ BCE are equilateral triangles. (1) prove: △ Dah ∽ ECH; (2) if ah: HB = 1:4, find s △ Dah: s △ ECH
- 19. It is known that, as shown in the figure, in △ ABC, the bisector of ∠ ABC and ∠ ACB intersects at point o
- 20. When the angle BAC is equal to 50, the angle BOC is equal to 50