# Find intersection point of 3 circles

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

• You mean, you know the coordinates of each Center? – imranfat Feb 1 '16 at 16:11
• Yes, I do mean the center of each circle. – Abu Bakr Feb 1 '16 at 16:21
• 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 – Jasser Feb 1 '16 at 16:25
• 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. – Paul Feb 1 '16 at 16:27

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.

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.

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

• 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. – Em. Jul 3 '16 at 10:17