## Tangent lines to circles - Construction

The legend on the right-hand side of the graphic serves not only to distinguish the tangents by color, but also as a switch with which the construction lines can be shown or hidden.

### The tangent to a circle k at a point B.

The basis for the construction is that the tangent of a circle runs perpendicular to the contact radius. So you may draw the straight line through M and B and construct the perpendicular straight line through B.

### The tangents to a circle k through a point P outside the circle

The points of contact B_{1} and B_{2} of the tangents are obtained as points of intersection of the Thales circle over the segment MP with the circle k.

The straight lines (PB_{1}) and (PB_{2}) are the desired tangents.

The additionally drawn straight line (B_{1} B_{2}) is the polar of P, which we used to calculate the tangent equations.

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

The points of contact B_{1} and B_{2} of the tangents are obtained as intersections of the perpendicular line from M to g with the circle k.

The parallels to g through B_{1} and B_{2} are the desired tangents.

### The common tangents to two circles k_{1} and k_{2}

In order to construct the outer tangents to two circles k_{1} (M_{1} , r_{1} ) and k_{2} (M_{2} , r_{2} ),
one first draws a circle k_{3} around M_{1} with radius r_{3}=r_{1}−r_{2}.

Its intersection points S_{1} and S_{2} with the Thales circle over M_{1}M_{2} are the points of contact of two tangents that can be placed from M_{2} to the circle k_{3}.

The half lines from M_{1} through S_{1} and S_{2} resp. intersect the circle k_{1} at the points of contact of the sought outer tangents. These are parallel to the tangents to the auxiliary circle k_{3}.

If k_{2} lies completely outside of k_{1}, there are also two "inner" tangents that cross between the circles.

In order to construct the inner tangents to two circles k_{1}(M_{1},r_{1}) and k_{2}(M_{2},r_{2}), one first draws a circle k_{3} around M_{1} with a radius r_{3}=r_{1}+r_{2}.

Its intersection points S_{1} and S_{2} with the Thales circle over M_{1}M_{2} are the contact points of two tangents that can be placed from M_{2} to circle k_{3}.

The half-straight lines from M_{1} through S_{1} and S_{2} resp. intersect the circle k_{1} at the points of contact of the inner tangents sought. These are parallel to the tangents to the auxiliary circle k_{3}.

See also:

Adjusting the Coordinate SystemsWikipedia: Tangent_lines to circles | Pole and polar