Hello, antero!

How to construct a circle which is tangent to two parallel lines and another circle?

The lines and circle are given but it is not said anything else about them.

Since the solution circle is tangent to the two parallel lines,

. . it lies between the two lines

. . and its center lies on the line halfway between them.

The given circle must have a portion between the two lines.

The diagram could look like this. Code:

L ------------*--------------------------------------
:
:
:r
:
:
C - - - - - - * - - - - - - - - - - - -
*
* : *
* : *
* :R *
M -----------------*------------:------------*---------
:
* : *
o
P

We are given parallel lines and

. . and a circle with center and radius

Construct the line halfway between and parallel to lines and

The center of our circle lies on line

The radius of this circle is the distance between lines and .

Spread your compass to a distance .

Using as center, swing an arc cutting line at point

Point is the center of our circle.

This diagram should be convincing . . .

Code:

L ----------*-*-*------------------------------------
* : *
* : *
* :r *
:
* : *
C - * - - - - o - - - - * - - - - - - -
* Q * *
* *
* r * * * : *
* o : *
* * * R :R *
M ----------*-*-*--*------*-----:------------*---------
* :
* * : *
o
P