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so first of all, I just want to point out that I am a beginner, so cut me some slack.

As the title says I have 3 circles. I know the coordinates of each center and the radius of each circle.

What I want to know is a formula that I can calculate the intersection point(points) with if any are present.

As the picture: 3 circles

Thank you for your help!!

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  • $\begingroup$ You mean, you know the coordinates of each Center? $\endgroup$ – imranfat Feb 1 '16 at 16:11
  • $\begingroup$ Yes, I do mean the center of each circle. $\endgroup$ – Abu Bakr Feb 1 '16 at 16:21
  • $\begingroup$ I think its better to find a formula to find intersection points of two circles each and then after getting the intersection points then find a common point which lie on each of these circles $\endgroup$ – Jasser Feb 1 '16 at 16:25
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    $\begingroup$ If $C_1$, $C_2$ and $C_3$ are the equations of your 3 circles then $C_1-C_2$, $C_2-C_3$ and $C_1-C_3$ are the equations of 3 straight lines in 2 variables. If these are consistent then their solution is your unique (triple) intersection point. $\endgroup$ – Paul Feb 1 '16 at 16:27
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Let each circle be defined by its centre $(x_i,y_i)$ and radius $r_i$.

The equation of a circle is given by $(x-x_i)^2+(y-y_i)^2=r^2_i$

So for two circles we have a pair of simultaneous equations:

They are: $x^2-2xx_1+x^2_1+y^2-2yy_1+y^2_1=r^2_1$

and $x^2-2xx_2+x^2_2+y^2-2yy_2+y^2_2=r^2_2$

Are you happy dealing with that? You find two points where the two circles intersect. Then test each one to see if it obeys the equation of the third.

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Generally you can represent circles with a center $(x_0,y_0)$ and a radius $r$ in the following form, using the pythagorean theorem: The points $(x,y)$ on this circle are exactly the points that satisfy

$$(x-x_0)^2 + (y-y_0)^2 = r^2$$

You can write down this equation for all three circles. By evaluating the difference between each pair of equations (note that $x^2$ and $y^2$ will cancel out), you get three lines that go throu the two intersection points of the corresponding pair of circles. Now you can just find the intersection of those lines.

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Let (x1,y1) , (x2,y2) and (x3,y3) be the centres of three circles. Assuming point of intersection(x,y) exists. image

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    $\begingroup$ You are really expected to be able to type up your answers. Questions posted as images are frowned upon. Answers posted as images are especially frowned upon. Formatting tips here. $\endgroup$ – Em. Jul 3 '16 at 10:17

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