# solving equation of circles

I have $2$ points on circle from which I want to find the center of circle. I also know the r. I am trying to solve $2$ equations but not able to find the centre $( C_x,C_y)$. Can someone check what I am doing wrong?

$1. (A_x - C_x)^2 + (A_y - C_y)^2 = r^2$

$2. (B_x - C_x)^2 + (B_y - C_y)^2 = r^2$

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...

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## 1 Answer

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.

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