## Tangents 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 to show or to hide the construction lines.

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

The basis for the construction is that the tangent of a circle is perpendicular to the contact radius. So you can 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 B1 and B2 of the tangents are obtained as the points of intersection of the Thales circle over the segment MP with the circle k.
The straight lines (PB1) and (PB2) are the desired tangents.

The additional line (B1 B2) 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 B1 and B2 of the tangents are obtained as intersections of the perpendicular line from M to g with the circle k.
The parallels to g through  B1  and  B2  are the desired tangents.

### The common tangents to two circles k1 and k2

In order to construct the outer tangents to two circles k1 (M1 , r1 ) and k2 (M2 , r2 ), one first draws a circle  k3  around  M1  with radius  r3=r1−r2.
Its intersection points  S1  and  S2  with the Thales circle over  M1M2  are the points of contact of two tangents that can be placed from  M2  to the circle  k3.
The half lines from  M1  through  S1  and  S2  resp. intersect the circle  k1  at the points of contact of the sought outer tangents. These are parallel to the tangents to the auxiliary circle  k3.

If  k2 is completely outside of  k1, there are also two "inner" tangents that cross between the circles.
To construct the inner tangents to the two circles  k1(M1,r1)  and  k2(M2,r2), first draw a circle  k3  around  M1  with a radius  r3=r1+r2.
Its intersection points  S1  and  S2  with the Thales circle over  M1M2  are the contact points of two tangents that can be placed from  M2  onto the circle  k3.
The half-straight lines from  M1  through  S1  and  S2 resp. intersect the circle  k1  at the points of contact of the inner tangents sought. These are parallel to the tangents to the auxiliary circle  k3.