## 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} = r^{2} .

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}) = r^{2} .

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 B_{1} and B_{2} calculated in this way, the equations of the tangents
t_{1} = (PB_{1}) and t_{2} = (PB_{2}) 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 B_{1} and B_{2} . 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 k_{1} and k_{2}

If r_{1} and r_{2} are the radii of the circles k_{1} and k_{2}, we assume w.l.o.g. that r_{1}>r_{2}.
If not, the circles are swapped.

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

|M_{1}M_{2}| < r_{1} − r_{2}

|M_{1}M_{2}| = r_{1} − r_{2}

r_{1} − r_{2} < |M_{1}M_{2}| < r_{1} + r_{2}

|M_{1}M_{2}| = r_{1} + r_{2}

|M_{1}M_{2}| > r_{1} + r_{2}

Special case r_{1} = r_{2}

a) |M_{1}M_{2}| = r_{1} − r_{2}

The easiest way to calculate the common contact point B is to add the r_{2}/|M_{1}M_{2}|-fold of the vector from M_{1} to M_{2}2 to the position vector of M_{2}. The tangent is the perpendicular to (M_{1}M_{2}) in B.

b) |M_{1}M_{2}| > r_{1} − r_{2}

In order to determine the outer tangents to two circles, one first determines the tangents from M_{2} to a circle k_{3} around M_{1}
with radius r_{3} = r_{1} − r_{2} (analogous to "Tangent to k through P"). If P_{1} and P_{2} are the contact points on k_{3},you may shift them by r_{1}/r_{3} times the vector from M_{1} to P_{1} or P_{2} resp. in order to get the contact points B_{1} and B_{2} (cf. a)).

Two circles with the same radius form a special case, since the auxiliary circle k_{3} would have a radius of zero. In this case the tangents are determined analogously to "Tangent to k parallel to g" with g = (M_{1}M_{2}).

c) |M_{1}M_{2}| = r_{1} + r_{2}

If the two circles touch from outside, there is a third common tangent t_{3} . The common point of contact divides the distance M_{1} M_{2} in the ratio r_{1}:r_{2}. The tangent t_{3} is orthogonal to M_{1}M_{2}.

d) |M_{1}M_{2}| > r_{1} + r_{2}

If k_{2} is completely outside k_{1}, 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 M_{2} to a circle k_{3} around M_{1}
with radius r_{3}=r_{1}+r_{2} (analogous to "Tangent to k through P").
If P_{1} and P_{2} are the contact points on k_{3}, then these are shifted by r_{1}/r_{3} times the vector
from M_{1} to P_{1} or P_{2} resp. (cf. a)).