Let three circles be located such that their centers are non-collinear.

Construct a fourth circle that is orthogonal to each of the three circles.

Let’s define our x-axis to connect the centers of two of the circles with the third circle to be toward the right of (or directly above) the leftmost circle which we shall designate its center to be the origin (0,0) in a Cartesian system.

The circle at the origin can be defined with the equation
x² + y² = r₁²
where r₁ is the circle’s radius.

The circle to its right with its center lying on the x-axis can be defined with the equation
(x – a)² + y² = r₂²
where r₂ is the circle’s radius and a is the distance its center is to the right of the y-axis and origin.

The third circle then can be defined with the equation
(x – b)² + (y – c)² = r₃²
where r₃ is the circle’s radius, b is the distance its center is to the right of the y-axis and c is the distance its center is above the x-axis.

For the fourth circle to be orthogonal to the first three, its radius and center should be where
r₄² + r₁² = d₁², r₄² + r₂² = d₂², and r₄² + r₃² = d₃²
where = d₁, d₂, and d₃ each are the corresponding distances between the fourth circle and the other circle of the respective radius, i.e., r₄² = d₁² – r₁² = d₂² – r₂² = d₃² – r₃².

----

Not sure exactly how to proceed, so, i shall just give one example of three circles orthogonal to the fourth....

Let’s assign a radius of 5 to the first circle. The first circle will then be defined by the equation x² + y² = 25. 

Let’s assign a radius of 9 to the second circle with its center at Cartesian coordinate (28, 0). The second circle will then be defined by the equation (x – 28)² + y² = 784.

Let’s assign a radius of 16 to the third circle with its center at Cartesian coordinate (13,20). The third circle will then be defined by the equation (x – 13)² + (y – 20)² = 400.

The fourth circle, orthogonal to the other three, will then have a radius of 12, its center at Cartesian coordinate (13,0) and be defined by the equation (x – 13)² + y² = 144.

Posted by Dej Mar on 2008-05-26 10:53:45