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In the triangle PMN with area of 1, Tan ∠ PMN = 1 / 2, Tan ∠ PNM = - 2, a suitable coordinate system is established, and the hyperbolic equation with m, n as the focus and passing through point P is obtained
If the height of Mn side is h, Mn = H / (Tan ∠ PMN) + H / (Tan ∠ PNM) = 3H / 2 (this formula is best derived by drawing a graph. First, it is understood by acute angle triangle, and then extended to obtuse angle triangle). The area of triangle PMN is s = Mn * H / 2 = 3H ^ 2 / 4 = 1, and H = 2 / sqrt (3), Mn = sqrt (3). (sqrt = root sign) so the ellipse
Given the angle MPN, ad is on PM, C and B are on PN, a and B intersect CD on F. if pf bisects the angle MPN, the proof is: 1 / PA + 1 / Pb = 1 / PC + 1 / PD
The three sides of △ PAB are cut by the straight line CD at C, F and D. according to Menelaus theorem: (AD / PD) * (BC / PC) * (FB / FA) = 1. Because pf bisection ∠ APB is obtained by angular bisection theorem: FB / FA = Pb / PA, the above two formulas are: AD / PA * PD = BC / Pb * PC, that is, (pd-pa) / PA * PD = (pb-pc) / Pb * PC, so 1 / pa - 1 / PD
Point O is on the bisector of angle MPN, and circle O intersects PN and pm at points a, B and C, D respectively
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RELATED INFORMATIONS
- 1. If the angle MPN is known, ad is on PM, C and B are on PN, and a and B intersect CD on F. (1) if pf bisects the angle MPN, the proof is: 1 / PA + 1 / Pb = 1 / PC + 1 / PD; (2) if 1 / PA + 1 / Pb = 1 / PC + 1 / PD, PF bisects the angle MPN
- 2. Ad bisector ∠ BAC, the vertical bisector of ad intersects the extension line of BC with point E, intersects ad with point F, and proves ∠ EAC = ∠ B Use the vertical bisector of a line segment
- 3. In the triangular pyramid p-abc, point O is perpendicular to the triangle ABC, and Po is perpendicular to the plane ABC
- 4. Is the dihedral angle range [0, π / 2], or is the plane angle range [0, π / 2)? Is a dihedral angle a corner? Does it have scope, What is the relationship with the plane angle range?
- 5. It is known that the angle between line AB and plane a is 30 degrees, the angle between line AC and plane a is 6O degrees, ab = 6cm, AC = 8cm, and the projective ab 'and AC' of oblique line AB and AC in plane a are perpendicular to each other?
- 6. The slope angle of a hillside is 30 degrees. If one kilometer goes along a straight road with 45 degrees to the slope, the slope will rise A. 250 with 2m b.250 with 3M c.250 with 6m d.500m
- 7. On a flank slope with an inclined angle of 60 degrees, climb along the road with an angle of 30 degrees to the horizontal line of the slope toe for 100 meters How many meters
- 8. For a car with a mass of M = 1000kg, the maximum speed V1 = 12m / s when driving on a straight road at rated power, and the maximum speed V2 = 8m / s when driving on a slope with an increase of 1m for every 20m. Assuming that the friction force on the car is equal under any condition, G is taken as 10m / S2, the following formula is obtained (1) The size of friction force; (2) What is the rated power of automobile engine; (3) The maximum speed a car can reach when going down the slope
- 9. As shown in the figure, the straight lines AB, CD and EF all pass through point O, and ab ⊥ CD and og bisect ∠ BOE. If ∠ EOG = 25 ∠ AOE, calculate the degrees of ∠ EOG, ∠ DOF and ∠ AOE
- 10. It is known that the line AB and CD intersect at O, OM ⊥ CD, the perpendicular point is O, OA bisects ∠ MOE, and ∠ BOD = 28 ° to find the degree of ∠ AOM ∠ COE ∠ BOE
- 11. Let the eccentricity root of the left focus of the ellipse be 3 / 3, and the length of the line passing through point F and perpendicular to x cut by the ellipse be 4 root 3 / 3 1 elliptic equation Let AB be the left and right vertices of the ellipse,
- 12. How to find the plane angle of dihedral angle?
- 13. Plane angle vertical plane method of dihedral angle What is the vertical plane method in the common method of making plane angle of dihedral angle?
- 14. The intersection of two planes forms the plane angle How many parts of the space are the three planes that do not coincide with each other? Some say 8, some say 7, and 8
- 15. In the dark house, there is a plane mirror on the table. Behind the mirror is a white wall. When you shine a flashlight on the mirror and the wall, the wall is very bright because the wall has a very good effect on the light______ The mirror is dark because it makes the light shine______ ．
- 16. It is known that the plane angle of dihedral angle α - ab - β is an acute angle, C is a point in plane α (it is not on edge AB), D is the projection of point C on plane β, and E is the projection of point C on edge ab
- 17. As shown in the figure, AB and CD are the two chords of circle O. points E and F are the midpoint of AB and CD, connecting EF, angle AEF = angle CFE, and proving AB = CD
- 18. Given that the coordinates of point P (x, y) satisfy the condition x + y ≤ 4Y ≥ XX ≥ 1, and point O is the origin of coordinates, then the minimum value of | Po | is equal to___ , the maximum value is equal to___ ．
- 19. (1 / 2) in the pyramid p-abcd, the bottom surface ABCD is rectangular, the vertical plane ABCD of PA, PA = ad = 1, ab = 2, e and F are the midpoint of AB and PD respectively（
- 20. As shown in the figure, AB is perpendicular to plane BCD, BC is perpendicular to CD, ab = BC, and the angle between AD and plane BCD is 30 degrees Finding the angle between AD and plane ABC Picture in