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Math Help - Point inside circle

  1. #1
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    Point inside circle

    Circle with radius r (r is an integer) has point P inside it. Center of circle is M(r,r).
    Points A,B,C are on the circumference: PA, PB and PC are different integers.
    AngleAPB = angleBPC = angleCPA = 120 degrees (of course!).

    I'm looking for hints/suggestions as to simplest way to "find" such circles,
    and get coordinates of point P. Anybody got "short cuts"? Thanks.
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  2. #2
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    Are you sure we can find such circle ?

    Let  x,y,z be the length of  AP , BP , CP respectively .

    and  a,b,c be the sides  BC,CA,AB respectively .

    Then we have

     a^2 = y^2 + z^2 + yz ,
     b^2 = z^2 + x^2 + zx ,
     c^2 = x^2 + y^2 + xy

    If they are set to be integers , it is not a problem . However , the radius of the circumcircle is :

     \frac{abc}{4S } = \frac{abc}{\sqrt{3} (xy+yz+zx) }

    which is irrational .

    EDIT : Oh ,  a,b,c are not necessarily integers , my mistake , sorry .
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  3. #3
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    Quote Originally Posted by simplependulum View Post
    EDIT : Oh ,  a,b,c are not necessarily integers , my mistake , sorry .
    Right.
    An example of such a circle: radius = 1729
    x=715, y=1690, z=2288 ; a=3458, b=2717, c=2139.0827....
    NOTE: BC (which is a) = diameter, but that's ok!
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