Can Three Circles Intersect at a Common Point?

In summary, the conversation discussed the intersection of three circles and whether it could be solved using a matrix or a standard equation. The equations for the three circles were provided and it was shown that the point (3, 1) is where all three circles intersect. Another potential intersection point was also found at (11/5, 13/5).
  • #1
karush
Gold Member
MHB
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studying with a friend there was the intersection of 3 circles problem which is in common usage
here is my overleaf output
View attachment 9075

I was wondering if this could be solved with a matrix in that it has squares in it

or is there a standard equation for finding the intersection of 3 circles given the centers and radius'
and an assumed intersection
 
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  • #2
Yes, the three circles have equations
[tex](x- 7)^2+ (y- 4)^2= 25[/tex]
[tex](x+ 9)^2+ (y+ 4)^2= 169[/tex] and
[tex](x+ 3)^2+ (y- 9)^2= 100[/tex]

Multiplying those squares gives
[tex]x^2- 14x+ 49+ y^2- 8y+ 16= 25[/tex]
[tex]x^2+ 18x+ 81+ y^2+ 8y+ 16= 169[/tex]
[tex]x^2+ 6x+ 9+ y^2- 18y+ 81= 100[/tex]

And subtracting will get rid of the squares!

Subtracting the first equation from the second gives
[tex]32x+ 32+ 16y= 144[/tex]
Subtracting the first equation from the third gives
[tex]20x- 40- 10y+ 65= 75.

32x+ 16y= 112 so 2x+ y= 7
20x- 10y= 50 so 2x- y= 5.
Adding those 4x= 12 so x= 3 and then y= 1.

That, (3, 1), is the point where all three circles intersect.

We also can look at 2x+ y= 7, so y= 7- 2x and [tex](x- 7)^2+ (y- 4)^2= (x- 7)^2+ (3- 2x)^2= x^2- 14x+ 49+ 9- 12x+ 4x^2= 5x^2- 26x+ 58= 25[/tex]. [tex]5x^2- 26x+ 33= 0[/tex]. That can be factored as [tex](5x- 11)(x- 3)= 0[tex] so x= 3 or x= 11/5. If x= 3 y= 7- 6= 1 and if x= 11/5, y= 7- 22/5= (35- 22)/5= 13/5. (3, 1) and (11/5, 13/5) is another intersection.
 

Related to Can Three Circles Intersect at a Common Point?

What is the "39 intersection of 3 circles" problem?

The "39 intersection of 3 circles" problem is a mathematical puzzle that involves finding the number of points of intersection when three circles intersect each other in a plane. It is also known as the "three-circle problem".

How is the solution to the "39 intersection of 3 circles" problem calculated?

The solution to the "39 intersection of 3 circles" problem is calculated using a formula known as the "Viviani's Theorem". This theorem states that the sum of the distances between the center of each circle and the point of intersection is equal to the sum of the radii of the three circles.

What is the significance of the "39" in the problem's name?

The number "39" in the name of the problem refers to the maximum number of points of intersection that can be formed when three circles intersect each other. This number is derived from the solution to the problem, which is 39.

What are some real-world applications of the "39 intersection of 3 circles" problem?

The "39 intersection of 3 circles" problem has various applications in fields such as geometry, engineering, and computer science. It is used in designing circular structures, analyzing traffic patterns, and creating computer graphics. It also has applications in the study of planetary orbits and the arrangement of atoms in molecules.

Are there any variations of the "39 intersection of 3 circles" problem?

Yes, there are variations of the "39 intersection of 3 circles" problem, such as the "n intersection of k circles" problem where n and k represent any number of circles. There are also variations that involve different shapes, such as the "intersection of 3 ellipses" problem. These variations may have different solutions and applications.

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