Let A B be an arc of a circle. Tangents are drawn at A and B to meet at C. Let M be the midpoint of arc A B. Tangent drawn at M meet A C and B C at D, E respectively. Evaluate lim A B → 0 Δ A B C Δ D E C I don't think evaluating the limit will be a problem. But how do I find the areas of the triangles and in what parameters?

Question

Let   A  B be an arc of a circle. Tangents are drawn at   A and   B to meet at   C. Let   M be the midpoint of arc   A  B. Tangent drawn at   M meet   A  C and   B  C at   D,   E respectively. Evaluate      lim          A      B      →      0                  Δ      A      B      C              Δ      D      E      C      I don&#039;t think evaluating the limit will be a problem. But how do I find the areas of the triangles and in what parameters?

knolsaadme · Accepted Answer

Let   A  (  r  cos  ⁡  θ  ,  r  sin  ⁡  θ  )  ,  B  (  r  cos  ⁡  ϕ  ,  r  sin  ⁡  ϕ  )  ∈  {  (  x  ,  y  )  ∣      x    2    +      y    2    =      r    2    } then:  C      (    r                  cos        ⁡                  (                                    ϕ              +              θ                        2                    )                            cos        ⁡                  (                                    ϕ              −              θ                        2                    )                      ,    r                  sin        ⁡                  (                                    ϕ              +              θ                        2                    )                            cos        ⁡                  (                                    θ              −              ϕ                        2                    )                      )      M      (    r    cos    ⁡          (                        θ          +          ϕ                2            )        ,    r    sin    ⁡          (                        θ          +          ϕ                2            )        )    And:      A    B    :  x  cos  ⁡      (                  θ        +        ϕ            2        )    +  y  sin  ⁡      (                  θ        +        ϕ            2        )    −  r  cos  ⁡      (                  θ        −        ϕ            2        )    =  0        D    M    E    :  x  cos  ⁡      (                  θ        +        ϕ            2        )    +  y  sin  ⁡      (                  θ        +        ϕ            2        )    −  r  =  0So:            Δ              A        B        C                    Δ              D        E        C              =            |                                                                  cos                2                            ⁡                              (                                                      θ                    +                    ϕ                                    2                                )                                                    cos              ⁡                              (                                                      θ                    −                    ϕ                                    2                                )                                              +                                                    sin                2                            ⁡                              (                                                      θ                    +                    ϕ                                    2                                )                                                    cos              ⁡                              (                                                      θ                    −                    ϕ                                    2                                )                                              −          cos          ⁡                      (                                          θ                −                ϕ                            2                        )                                                                              cos                2                            ⁡                              (                                                      θ                    +                    ϕ                                    2                                )                                                    cos              ⁡                              (                                                      θ                    −                    ϕ                                    2                                )                                              +                                                    sin                2                            ⁡                              (                                                      θ                    +                    ϕ                                    2                                )                                                    cos              ⁡                              (                                                      θ                    −                    ϕ                                    2                                )                                              −          1                    |        2    =  4      cos    4    ⁡            θ      −      ϕ        4  Now as       A    B    →  0 or   θ  −  ϕ  →  0            lim              A        B        →        0                            Δ                  A          B          C                            Δ                  D          E          C                      =    4

pokoljitef2 · Accepted Answer

Hint: draw a circle of radius 1 around the origin. You can assume that   A  B is symmetrical w.r.t.   Y-axis. Let   A  =  (  cos  ⁡  (  α  )  ,  sin  ⁡  (  α  )  ) be in the first quadrant,   B  =  (  −  cos  ⁡  (  α  )  ,  sin  ⁡  (  α  )  ) with   0  &amp;lt;  α  ≤      1    2    π. Then       |    Δ  A  B  C      |    =  cos  ⁡  (  α  )  sin  ⁡  (  α  ) and       |    Δ  D  E  C      |    =            (      2      sin      ⁡      (      α      )      −      1              )        2                    tan      ⁡      (      α      )      . Now take your limit where   α  →      1    2    π, which should give you 1. Generalize this to a circle with radius   r.

Let A B be an arc of a circle. Tangents are drawn at A and B to meet at C . Let

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