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Math Help - have two points and there distance from third, find third

  1. #1
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    have two points and there distance from third, find third

    I have 2 points, and i know those two points distance from the location im trying to find. From that data i need to find a 3rd points location.

    Such as 6,8 is 5 from the location, and 0.0 is also 5 from the location. So it could be at 3,4.


    I just cant figure out the math to do it. Ive been trying to derive it from the distance formula but its not working.

    Thanks for any suggestions or help.
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  2. #2
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    I'd use the formula for a circle centred at a point (x_0,y_0) and radius = r

    (x-x_0)^2 + (y-y_0)^2 = r^2

    In this case you have two equations

    (x-6)^2 + (y-8)^2 = 25, \quad x^2 + y^2 = 25

    Solving simultaneously you get

    3x + 4y = 25 \qquad (1)

    Now substitute y = \frac{25 - 3x}{4} into the second circle equation you get

    x^2 + \left(\frac{25 - 3x}{4}\right)^2 = 25

    which simplifies to (x-3)^2 = 0 and therefore x = 3.

    Substituting back into equation (1) gives y=4.

    So your intuition was right but following the steps above you can prove (3,4) is the only solution.

    Generally for these sorts of problems there are either 2 distinct solutions, one exact solution (this case) or no solution. Infinitely many solutions are a possibility too but only when the circle centres and radii are equal.
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  3. #3
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    Thanks! im going to try and derive a useful java function from that for the program im trying to make. I fear that knowing a formula is only half the battle, and that formula is kinda monstrous :-) Well thanks for your help, im gonna jump into this soon.
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  4. #4
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    If you have the distance between the first two points you could probably solve it with trigonometry.
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