Ruler and Compass Construction 23
Circle of Given Radius Inscribed in a Given Angle

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1. Extend DC
2. Semi-circle centered at C, radius AB
3. Arc centered at F, arbitrary radius
4. Arc centered at G, same radius
5. Segment CH
6. Arc centered at F, radius AB
7. Arc centered at I, radius AB
8. Intersection K
9. Arc centered at G, arbitrary radius
10. Arc centered at K, same radius
11. Ray CL, bisector of the angle
12. Line JI, with intersection M
13. Circle centered at M, radius AB
Why it works: The center of the inscribed circle must lie on the angle bisector, and it must lie at distance AB from each side. So the angle bisector construction is used to get one line through the circle center, and the previous construction is used to get a line at distance AB from side CD. The intersection of these two lines is the center of the inscribed circle, and we already know the radius is AB.
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