Tangent lines to circles - Calculation

Pole and Polar

The circle k around the center M with radius r is described by the equation
(x - x_M)² + (y - y_M)² = r² .

If you insert the coordinates of point P into one of the factors in the squares on the left, you get the following equation, which describes a straight line.
(x_P - x_M)·(x - x_M) + (y_P - y_M)·(y - y_M) = r² .
It is called the polar to the pole P and is a powerful tool for finding the tangent equations.

The tangents to a circle k through a point P.

Given a circle k and a point P on the circle, the polar of P is the desired tangent

Given a circle k and a point P outside of k, the polar of P intersects the circle at the two points of contact of the tangents through P. The equation of the polar is solved for y (or for x if the coefficient of y is zero) and inserted into the circular equation. With the contact points B₁ and B₂ calculated in this way, the equations of the tangents t₁ = (PB₁) and t₂ = (PB₂) are set up.

If the point P is inside the circle, the polar is a straight line outside the circle. The points on it all have the property that their polar intersect in P.

The tangents to a circle k parallel to a straight line g

Given a circle k and a straight line g: a x + b y = c , the perpendicular to g through the center M of the circle intersects it at the contact points B₁ and B₂ . For the coefficients of the perpendicular line applies a '= - b and b' = a. The constant summand c' is obtained by inserting the coefficients of M.

Tangents to two circles k₁ and k₂

If r₁ and r₂ are the radii of the circles k₁ and k₂, we assume w.l.o.g. that r₁>r₂. If not, the circles are swapped.

How many common tangents two circles have depends on the mutual position of the circles:

a)  |M₁M₂| = r₁ − r₂
The easiest way to calculate the common contact point B is to add the r₂/|M₁M₂|-fold of the vector from M₁ to M₂2 to the position vector of M₂. The tangent is the perpendicular to (M₁M₂) in B.

b) |M₁M₂| > r₁ − r₂
In order to determine the outer tangents to two circles, one first determines the tangents from  M₂  to a circle k₃ around M₁ with radius r₃ = r₁ − r₂ (analogous to "Tangent to k through P"). If P₁ and P₂ are the contact points on k₃,you may shift them by r₁/r₃ times the vector from M₁ to P₁ or P₂ resp. in order to get the contact points B₁ and B₂ (cf. a)).

Two circles with the same radius form a special case, since the auxiliary circle k₃ would have a radius of zero. In this case the tangents are determined analogously to "Tangent to k parallel to g" with g = (M₁M₂).

c) |M₁M₂| = r₁ + r₂
If the two circles touch from outside, there is a third common tangent t₃ . The common point of contact divides the distance M₁ M₂ in the ratio r₁:r₂. The tangent t₃ is orthogonal to M₁M₂.

d) |M₁M₂| > r₁ + r₂
If k₂ is completely outside k₁, there are two "inner" common tangents that cross between the circles.
In order to determine the inner tangents to two circles, one first determines the tangents from  M₂  to a circle k₃ around M₁ with radius  r₃=r₁+r₂ (analogous to "Tangent to k through P"). If P₁ and P₂ are the contact points on k₃, then these are shifted by r₁/r₃ times the vector from M₁ to P₁ or P₂ resp. (cf. a)).

See also:

Wikipedia: Tangent_lines to circles | Pole and polar