1. ## 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.

2. Are you sure we can find such circle ?

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

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

Then we have

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

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

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

which is irrational .

EDIT : Oh , $\displaystyle a,b,c$ are not necessarily integers , my mistake , sorry .

3. Originally Posted by simplependulum
EDIT : Oh , $\displaystyle a,b,c$ 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!