How to find center of a circle given 2 points on the circle and the radius?

Question

I would like to find an equation that I can put into excel to calculate the coordinates for the center of the circle.

Given

$P_1\equiv (a=33855.05, b=21129.55)$

$P_2\equiv (c=33745.04, d=21221.69)$

$\text{Radius}= 590$

I know that the center lies on the perpendicular bisector between the two points, but don't know how to calculate it. What I have so far:

I calculated the Midpoint ($M$) of the chord
$M \equiv(e, f) \equiv\left(\frac{a+c}{2}, \frac{b+d}{2}\right) \equiv (e=338855.05, f= 21175.62)$

I also know the slope $m = \frac{b-d}{c-a} = 0.838$

So how do I find the coordinates of center? I am aware that there will be two possible answers.

Answer 1

Let $P(a,b)$ and $Q(c,d)$ be two given points known to be on a circle. Compute the midpoint $M$ of $\overline{PQ}$. Then compute the equation of the line through $M$ which is perpendicular to $\overline{PQ}$, i.e. find the equation of the perpendicular bisector of $\overline{PQ}$. The center $C(x_0,y_0)$ lies on the line just found, producing one equation in the unknowns $x_0,y_0$. On the other hand, $\overline{CP}=\overline{CQ}$ so the distance formula produces a second equation in the unknowns $x_0,y_0$. Solve these two equations to determine the center $C(x_0,y_0)$.

PS I just found this which might prove helpful.

Answer 2

I assume that you computation is correct.

Continuing from your work, you need to find the equation of the line of centers.

The line of centers is perpendicular to P1P2, and hence its slope M is -1/0.838.

Its equation, by slope-point form, is then y – f = M(x – e) ….. (1)

Next, you need to find the equation of the red circle [centered at P2(c, d) and radius = 590]. (See figure below.)

And it is:- $(x – c)^2 + (y – d)^2 = 590^2$ ….. (2)

Solving (1) and (2) will give you two sets of (x, y) correspond to the two candidates of the required center(s)