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Math Help - Circle from two points and an angle

  1. #1
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    Smile Circle from two points and an angle

    I want to determine the circle through points N, S and V.
    I know the distance between N and S.
    I know the angle NVS.

    How could an expression describing such a circle be constrained given that information?
    Thank you!
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  2. #2
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    Re: Circle from two points and an angle

    Quote Originally Posted by Encircled View Post
    I want to determine the circle through points N, S and V.
    I know the distance between N and S.
    I know the angle NVS.

    How could an expression describing such a circle be constrained given that information?
    Thank you!
    Hej,

    if - and only if - the 3 points are placed as in the attached sketch then

    \angle(NVS) = (x+y)^\circ

    The red lines are the radii which form 2 isosceles triangles which can be split into right triangles.

    The angle c at the center is calculated by:

    (180^\circ-2x^\circ) + (180^\circ-2y^\circ) = c^\circ

    The triangle \Delta(NSC) is an isosceles triangle.

    so r = \frac{\frac12 |\overline{NS}|}{\sin\left(\frac12 c^\circ\right)}
    Attached Thumbnails Attached Thumbnails Circle from two points and an angle-dreipktwinklkreis.png  
    Thanks from Encircled
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  3. #3
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    Re: Circle from two points and an angle

    Great answer, and fast, thanks alot!
    I can puzzle together the rest of what I need from this point. Magnificent.
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  4. #4
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    Re: Circle from two points and an angle

    Hello, Encircled!

    I want to determine the circle through points N, S and V.
    I know the distance between N and S.
    I know the angle NVS.

    How could an expression describing such a circle be constrained given that information?
    Code:
                  * * *
              *           *
            *  d/2  M  d/2  *
         N o- - - - + - - - -o S
            \ *     |     * /
          *  \   * θ|θ *r  /  *
          *   \     o     /   *
          *    \    O    /    *
                \       /
           *     \     /     *
            *     \   /     *
              *    \θ/    *
                  * o *
                    V
    We have a circle with center O.
    Chord N\!S = d.
    . . Midpoint M\!:\;MS = \tfrac{d}{2}
    Angle NV\!S = \theta.

    Draw radii ON = OS = r.
    \angle NOS = 2\theta;\;\angle MON = \angle MOS = \theta.

    In \Delta SMO\!:\;\sin\theta \,=\,\dfrac{\frac{d}{2}}{r} \:=\:\frac{d}{2r}

    Therefore: . r \:=\:\frac{d}{2\sin\theta}
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  5. #5
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    Re: Circle from two points and an angle

    I rephrased the question here:
    Want radius of circle through 3 points. A chord and an angle are known.
    (Because at first I actually failed to find this, my own, old thread, duh)
    Please see it in order to maybe clearer understand what I'm looking for.


    And actually, now when I look at it, there are some misunderstandings in this thread here.

    earboth:
    I do not know angles x or y. I only know the angle NVS = x+y.
    So I can't solve for radius according to your formula there.

    Soroban:
    Why would angles NOS = 2 NVS, and MOS = NVS?
    It looks wrong.
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  6. #6
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    Re: Circle from two points and an angle

    Quote Originally Posted by Encircled View Post
    ...

    And actually, now when I look at it, there are some misunderstandings in this thread here.

    earboth:
    I do not know angles x or y. I only know the angle NVS = x+y.
    So I can't solve for radius according to your formula there.

    ...
    That isn't necessary. I wrote you:

     (180^\circ-2x^\circ) + (180^\circ-2y^\circ) = c^\circ

     360^\circ-2x^\circ-2y^\circ = c^\circ

     360^\circ-2\underbrace{(x^\circ+y^\circ)}_{ \angle(NVS)}=c^\circ And now you don't need x and y anymore because you only uses the sum of them and this sum is known.
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  7. #7
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    Re: Circle from two points and an angle

    I borrow this fine ASCII art and introduce in it the point L.
    Now, since L is on the opposite side of the chord, as seen from the circle origo O, does it still hold that the angle NLS = half the angle NOS?
    It is supposed to hold for any point V on the same circle which the chord crosses, but intuitively, that angle seems much larger when V is at L on the smaller side of a chord. Is another relationship at play there?

    Code:
                  *L *
              *           *
            *  d/2  M  d/2  *
         N o- - - - + - - - -o S
            \ *     |     * /
          *  \   * θ|θ *r  /  *
          *   \     o     /   *
          *    \    O    /    *
                \       /
           *     \     /     *
            *     \   /     *
              *    \θ/    *
                  * o *
                    V
    Last edited by Encircled; January 25th 2013 at 05:12 AM.
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