Let plane α ∩ plane β = straight line L. ABC is three points and a ∈ α, B ∈ α, C ∈ β. The straight line AB is not parallel to L, and plane ABC ∩ β = M. judge the position relationship between L and m, and prove your conclusion

Let plane α ∩ plane β = straight line L. ABC is three points and a ∈ α, B ∈ α, C ∈ β. The straight line AB is not parallel to L, and plane ABC ∩ β = M. judge the position relationship between L and m, and prove your conclusion

Classification discussion:
1. If point C is on line L, plane ABC and plane α coincide, then line L coincides with line M
2. If point C is not on line L
∩ plane α ∩ plane ABC = AB, plane ABC ∩ plane β = m
∴AB‖m
∵ AB is not parallel to L
That M is not parallel to L
And ∵ M C plane β, L C plane β
∩ m ∩ L, m and l intersect in the plane β

Two mathematical problems in Senior Two
1. It is known that in △ ABC, angles a, B and C form an arithmetic sequence. Proof: 1 / (a + b) + 1 / (B + C) = 3 / (a + B + C)
2. We know that Tan α + sin α = a, Tan α - sin α = B. prove (A & sup2; - B & sup2;) & sup2; = 16ab

1.
Because the angles a, B, C are arithmetic
So the angle B = 60 degrees
To prove that 1 / (a + b) + 1 / (B + C) = 3 / (a + B + C)
Just prove B ^ 2 = a ^ 2 + C ^ 2-ac
According to the cosine theorem B ^ 2 = a ^ 2 + C ^ 2-2ac * CoSb
So B ^ 2 = a ^ 2 + C ^ 2-ac
So 1 / (a + b) + 1 / (B + C) = 3 / (a + B + C)
two
a²-b²=4tanαsinα
ab=tan^2 α-sin^2 α=tan^2 α(1-cos^2 α)=(tanαsinα)^2
Yi Zheng left = right

Let the intersection of the line L and the square of the circle C: x + the square of y = the square of R at two points a and B, o be the origin of the coordinate, and a (root 3.1) is known. When the distance from the origin o to the line L is root 3, the equation of the line L is solved

Substituting a (√ 3,1) into x ^ + y ^ = R ^, r = 2,
It is known that OA = ob = 2,
Let O and ab intersect C, OC = √ 3,
From the graph CB = CA = 1,
AB = OA = ob = 2,
So the angle AOB is 60 degrees and ab is parallel to the x-axis,
Get B coordinate (- √ 3,1)
The line L is y = 1