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 summary

Tangent lines circle locations

Tangent lines of equal radii

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

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:

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 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

Step 1: Calculating the intersection point of the two tangent lines:

The distance between the circle’s centers D is:

The outer tangent lines intersection point (x_p , y_p ) (r₀ > r₁) is:

Step 2: Once x_p and y_p 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 y_t1 ↔ y_t2.

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

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 at 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

Tangent lines between two equal radii circles

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:

⟶

From simple trigonometry we get the outer tangent points for both circles (see figure - 2).

x_1,2 = x₁ ± r sinθ	y_1,2 = y₁ ∓ r cosθ
x_3,4 = x₂ ± r sinθ	y_3,4 = 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: