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Triangle ABC is an equilateral triangle, PA is perpendicular to plane ABC, and D is the midpoint of BC
Because ABC is an equilateral triangle and D is the midpoint of BC, ad is perpendicular to BC
And because PA is perpendicular to plane ABC, PA is perpendicular to plane BC
Because PA intersects ad, BC is perpendicular to pad
It is known that P is a point outside the plane of triangle ABC, PC is perpendicular to AB, PC = AB = 2, e and F are the midpoint of PA and BC respectively
(1) Verification: EF and PC are straight lines in different planes;
(2) Find the angle between EF and PC
The position relationship between planar PCB and straight line EF: point F is a point on planar PCB, e is a point outside planar PCB, and straight line PC is on planar PCB. So straight line EF and PC are out of plane. (2) through e, eg is vertical AC, perpendicular foot is g, connecting GF. Because PC is vertical to planar ABC, planar PAC is vertical to planar ABC, eg is vertical
It is known that D and E are the points on the sides AB and AC of equilateral △ ABC, and BD = CE
It is known that D and E are the points of AB and AC in equilateral △ ABC, and BD = CE
Prove that △ ade is an equilateral triangle
(no picture)
It is proved that △ ABC is an equilateral triangle, so AB = AC
AD=AB-BD,AE=AC-CD
Because BD = CE
So ad = AE
Delta ade is an isosceles triangle
So ∠ ade = ∠ AED
And because ∠ a = 60
Therefore, ADE + AED = 2 ade = 180 - a = 120
Therefore, ADE = AED = 60
The three internal angles of △ ade are all 60 degrees, so they are equilateral triangles
As shown in the figure, on the sides AB and AC of equilateral triangle ABC, BD is equal to CE triangle, is ade equilateral triangle?
Yes
AB = AC, BD = CE = > ab-bd = ac-ce = > ad = AE, ∵ a = 60 ° = > Δ ade is an equilateral triangle
RELATED INFORMATIONS
- 1. In equilateral triangle ABC, is de / / BC. Angle ade an equilateral triangle?
- 2. 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
- 3. 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
- 4. 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
- 5. 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?
- 6. In RT △ ABC, ∠ C = 90 degree, ∠ a = 30 degree, then BC: AC: ab = what
- 7. Δ ABC ≌ Δ CDA, AB and CD, BC and DA are corresponding edges. Write other corresponding edges and corresponding angles
- 8. 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
- 9. 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 ′
- 10. 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
- 11. 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?
- 12. 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
- 13. If vector AP = 1 / 3 vector Pb and vector AB = Λ vector BP, then the value of Λ is?
- 14. Proof: as shown in Figure 9, in the triangle ABC, ab = AC,
- 15. As shown in the figure, in △ ABC, ∠ a = 36 °, ABC = 40 °, be bisection ∠ ABC, ∠ e = 18 °, CE bisection ∠ ACD? Why?
- 16. 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
- 17. It is known that, as shown in the figure, in △ ABC, the bisector of ∠ ABC and ∠ ACB intersects at point o
- 18. When the angle BAC is equal to 50, the angle BOC is equal to 50
- 19. In the triangle ABC, the angle a = 80 degrees, e, F, P are a point on the edge of AB, AC, BC respectively, be = BP, CP = CF to calculate the angle EPF
- 20. In ABC, AB is equal to AC, BF is the height on the side of AC, and the angle FBC is equal to 1 / 2 angle BAC