Tangent lines between two circles calculator Print if a point is inside or outside of a circle summary ▲
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 Print tangent lines between two circles summary ▲
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 operation 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
solved example
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 circle’s 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
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
Scheme of outer tangent 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 r1
correctness 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 Print number of tangentLines summary ▲
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
Tangent lines between two equal radii circles Print tangent lines between two circles summary ▲
Tangent lines between two equal radii
Figure - 1
Tangent lines between two equal radii detail
Figure - 2
The circles are given by the equations:
(x − x1) + (y − y1) = r2
(x − x2) + (y − y2) = r2
Because the radii are equal the outer tangent lines are parallel and the distance between them is  2r.
The equation of the line connecting the two circles center is:
y=(y_2-y_1)/(x_2-x_1 ) x+(x_1 (y_2-y_1 ))/(x_2-x_1 ) x_1
The slope angle is:
tan⁡〖α=m=(y_2-y_1)/(x_2-x_1 ) 〗 ⟶ α=tan^(-1)⁡〖(y_2-y_1)/(x_2-x_1 )〗
From simple trigonometry we get the outer tangent points for both circles (see figure - 2).
x1,2 = x1 ± r sinθy1,2 = y1 ∓ r cosθ
x3,4 = x2 ± r sinθy3,4 = y2 ∓ r cosθ
Note the corresponding ± and ∓ for the x and y coordinate.
The coordinate of the intersection point between the two inner tangent lines is:
x_d=(x_2-x_1)/2 y_d=(y_1-y_2)/2