Ruler and Compass Construction 32
Circle Tangent to Given Parallel Lines and Given Circle

1. Arbitrary perpendicular to the parallel lines
2. Midpoint of BC
3. Perpendicularto BC through D
4. Arbitrary ray from A
5. Add BD to AE to get F
6. Arc centered at A, through F
7. Circle centered at G, radius BD
Explanation of the construction: We know the center of the tangent circle has to be on the midline between the parallel lines, which also gives the radius of the circle. Then adding this radius to the given circle radius gives another locus for the center of the tangent circle, much like construction 28.
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