Tangent lines between two circles calculator
Circles of the form:     (x − a)2 + (y − b)2 = r2
Circle 1: ( x − )2 + ( y − )2 = 2
Circle 2: ( x − )2 + ( y − )2 = 2
Circles of the form:     x2 + y2 + Ax + By + C = 0
Circle 1: x2 + y2 + x + y + = 0
Circle 2: x2 + y2 + x + y + = 0
Intersection of outer tangent lines:
Intersection of inner tangent lines:
Number of tangent lines:
Distance between the circles centers:
Outer lines tangent points:
Inner lines tangent points:
Outer 1st tangent line equation:
Outer 2nd tangent line equation:
Inner 1st tangent line equation:
Inner 2nd tangent line equation:
        
    
Tangent lines between two circles
Equation of the two circles given by:
(x − a)2 + (y − b)2 = r02
(x − c)2 + (y − d)2 = r12
Scheme of two circles tangent lines
Example:   Find the outer intersection point of the circles:
(r0)       (x − 3)2 + (y + 5)2 = 42
(r1)       (x + 2)2 + (y − 2)2 = 12
The intersection point of the outer tangents lines is:   (-3.67 ,4.33)
Note: r0 should be the bigger radius in the equation of the intersection.
r0 = 4         a = 3         b = − 5
The tangent points on   r0   are:
xt1 = 5.24       and       xt2 = − 0.86
yt1 = − 1.69       and       yt2 = − 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
Scheme of outer tangent lines
Step 1:   Calculating the intersection point of the two tangent lines:
The distance between the circles centers D is:
Distance circles centers
The outer tangent lines intersection point (xp , yp )   (r0 > r1) is:
Step 2:   Once xp and yp were found the tangent points of circle
radius r0 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     yt1 ↔   yt2.
Step 3:   Finding the outer tangent points of circle   r1
Correctnass 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
Scheme of outer tangent lines
Step 1:   Calculating the intersection point of the two tangent lines:
Step 2:   Same as before.
Correctnass check if required (s should be equal to 1):
Step 3:   Finding the inner tangent points on circle r1
Correctnass check if required (s should be equal to 1):
Step 4:   The lines equations of the outer tangents lines are:
Number of tangent lines in different circles location
Scheme Number of tangent lines
Circles example
Condition
0
x2 + y2 = 42
(x − 1)2 + y2 = 22
D - is the distance between circles centers
D < |r0 − r1|
1
(x − 2)2 + (y − 4)2 = 42
(x − 3)2 + (y − 4)2 = 32
D = |r0 − r1|
2
(x + 2)2 + (y + 2)2 = 42
(x − 2)2 + (y − 4)2 = 52
|r0 − r1 | < D < r0 + r1
3
x2 + y2 = 42
x2 + (y − 7)2 = 32
D = r0 + r1
4
x2 + y2 = 42
(x − 6)2 + (y − 5)2 = 32
D > r0 + r1