Tangent points from a point (x_{p} , y_{p}) on a circle.
Example: Find the tangent points on the circle (x − 2)^{2} + (y + 5)^{2} = 9 from point (7 , 1).
In this case: a = 2 b = − 5 r = 3
The value of the square root is:
x_{1} = 4.87 x_{2} = 0.61
y_{1} = − 5.89 y_{2} = − 2.34
So first tangency point is: (4.87,5.89) and the second point is the other points: (0.61,2.34)
Now we can check if the tangent point that we found is on the circle:
(4.8662)^{2} + (5.888 + 5)^{2} =
2.866^{2} + (.888)^{2} = 9
Note: we used higher precision of the point coordinate otherwise we would get slightly different value then 9.
nomenclature:
D −  Distance from point to circle center 
d −  Distance from point to tangent point 
θ −  Angle between the two tangent lines 
x_{1,2}  Tangent points x coordinates 
y_{1,2}  Tangent points y coordinates 
x_{i}  line connecting point to circle center x intercept 
y_{i}  line connecting point to circle center y intercept 

The distance between the point (x_{p} , y_{p}) and the tangent point (1) is:
The angle between the two tangent lines θ is:
Note: in the equations above x_{1} can be replaced by x_{2}.
Circle form: 
x^{2} + y^{2} = r^{2} 
Line connecting point (x_{p} , y_{p}) with circle center 
equation: 

x intercept: 
x_{i} = 0 
y intercept: 
y_{i} = 0 

Circle form: 
(x − a)^{2} + (y − b)^{2} = r^{2} 
Line connecting point (x_{p} , y_{p}) with circle center 
equation: 

x intercept: 

y intercept: 


Circle form: 
x^{2} + y^{2} + Ax + By + C = 0 
Line connecting point (x_{p} , y_{p}) with circle center 
equation: 

x intercept: 

y intercept: 


