Ruler and Compass Construction 39
Apollonius' Problem: Circle tangent to three circles

Reset. Line LM, circle BC, point A.
1. Line through B, perpendicular to LM.
2. Midpoint F between circle and line.
3. Line FZ through F parallel to LM.
4. Expanded circles, centered at B and L, meet at F.
5. Circle OA expanded by same amount.
6. Inversion circle centered at F, arbitrary radius.
7. Inversion of circle BF to line HI, and LF to HH-II.
8. Ray FO, with intersections J,K.
9. Inversions of J and K to P and Q
10. Midpoint R of P and Q is center of inverted circle.
11. Midline between lines HI and HH-II.
12. Circle centered at R, radius sum of PR and FY.
13. Circle centered at S,radius FY, tangent to circle RP.
14. Ray FS, with intersections T,U.
15. Inversions V,W of T,U.
16. Midpoint X of T,U.
17. Circle centered at X radius distance to O less radius OA. Tangent to original line and circles.