Hello everyone I have 2 points A and B.
And I have to proof that for all X points for them $\frac{AX}{BX} = 3$ they are on a one circle.
I tried to convert the equation to something like $x^2 + y^2 = 3$
and calculate the length of AX and BX and place in the equation but I didn't success.
Equivalently,
$$AX^2=9BX^2$$ is
$$(x-x_A)^2+(y-y_A)^2=9((x-x_B)^2+(y-y_B)^2)$$
or
$$x^2-\frac{9x_B-x_A}4x+\frac{9x_B^2-x_A^2}8+y^2-\frac{9y_B-y_A}4y+\frac{9y_B^2-y_A^2}8=0,$$
which is the equation of a circle.