Problem 5
Two tangent circles and a point P on their common tangent perpendicular to the line joining their centres are given. Construct with ruler and compass all circles tangent to the two given circles and passing through P.
Step 5 of 5: Verify completeness
Detailed analysis
Every required circle maps to a common tangent different from PT, and there are exactly two such tangents for externally tangent circles. The construction therefore produces all and only the two required circles.