I have two points on a circle from which I want to find the center of the circle. I also know the radius $r$. I am trying to solve two equations but not able to find the centre $(C_x,C_y)$. Can someone check what I am doing wrong?
$(A_x - C_x)^2 + (A_y - C_y)^2 = r^2$
$(B_x - C_x)^2 + (B_y - C_y)^2 = r^2$
Solving 1 first, I got
$-C_x^2 = r^2 - A_x^2 - A_y^2 + C_y^2$
Putting this value in eq. 2, I got nothing ... how to find $C_x$ and $C_y$ then?
$B_x^2 - A_x^2 - A_y^2 + B_y^2 = 0$
these are the values I already know...
The initial equations you wrote are correct. The problem is incorrectly squaring.
$$(a+b)^2=(a+b)(a+b)=a(a+b)+b(a+b)=a^2+ab+ba+b^2=a^2+2ab+b^2.$$
Similarly,
$$(a-b)^2=(a-b)(a-b)=a^2-2ab+b^2.$$
So, for example, $(A_x-C_x)^2=A_x^2-2A_xC_x+C_x^2$, not $A_x^2-C_x^2$.
If you subtract the equations, you will be left with a linear equation relating $C_x$ and $C_y$. You could solve for one in terms of the other, then substitute back into one of the original equations.