Ruler and Compass Construction 33
Circle Tangent to Given Circle On Angle Bisector

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1. Intersection E of circle and angle bisector
2. Perpendicular to AB through E
3. Circle centered at E through F
4. Intersection G of the circle with angle bisector
5. Segment FG
6. Line parallel to FG through E
7. Perpendicular to AB through H, intersection I with bisector
8. Circle centered at I, through H

Explanation of the construction: We know point E will be a tangent point of the
desired circle. So I construct a circle at E tangent to the angle sides, and use
similarity to find the tangent circle center I. Transversals AD adn AB are cut
in equal ratios by parallels HE and FG.