Equation of the two circles given by:
(x − a)^{2} + (y − b)^{2} = r_{0}^{2}
(x − c)^{2} + (y − d)^{2} = r_{1}^{2}
Example: Find the outer intersection point of the circles:
(r_{0}) (x − 3)^{2} + (y + 5)^{2} = 4^{2}
(r_{1}) (x + 2)^{2} + (y − 2)^{2} = 1^{2}
The intersection point of the outer tangents lines is: (3.67 ,4.33)
Note: r_{0} should be the bigger radius in the equation of the intersection.
r_{0} = 4 a = 3 b = − 5
The tangent points on r_{0} are:
x_{t1} = 5.24 and x_{t2} = − 0.86
y_{t1} = − 1.69 and y_{t2} = − 6.04
For demonstration purpose we took the wrong pair of points (5.24, 6.04) and check the value of s (this opertion is
not required if the correct signs are applied).
Because s≠1 swap between y values to get the point: (5,24 , 1.69)
First tangent point is at:
(5.24 , − 1.69)
Second tangent point at:
(− 0.86 , − 6.04)
For a complete example see

Calculating the outer tangents lines
Step 1: Calculating the intersection point of the two tangent lines:
The distance between the circles centers D is:
Step 2: Once x_{p} and y_{p} were found the tangent points of circle
radius r_{0} can be calculated by the equations:
Note: it is important to take the signs of the square root as positive for x and negative for y or vice versa,
otherwise the tangent point is not the correct point. It is possible to check the correctness of the point by
calculating the value of s in the following formula, if s = 1 then the point is correct otherwise swap
the y values y_{t1} ↔ y_{t2}.
Step 3: Finding the outer tangent points of circle r_{1}
correctness check if required (s should be equal to 1):
Step 4: The lines equations of the outer tangents lines are:
Calculating the inner tangents lines
Step 1: Calculating the intersection point of the two tangent lines:
Step 2: Same as before.
correctness check if required (s should be equal to 1):
Step 3: Finding the inner tangent points on circle r_{1}
correctness check if required (s should be equal to 1):
Step 4: The lines equations of the outer tangents lines are:
