Ruler and Compass Construction 34
Circle Tangent Inside Given Angle and Through Given Point

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1. Perpedicular to AB through D
2. Circle centered at A, radius DE
3. Perpendicular to AB through F
4. Angle bisector
5. Circle centered at E, radius FG
6. Segment AD
7. Perpendicular to AD at D, intersection F
8. Circle centered at A, through H
9. Circle centered at H, radius DI
10. Perpendicular to AB at J
11. Circle centered at K, through D
Explanation of the construction: Let the slope of the angle bisector be m, let D = (a,b), and let (x,mx) be the center of the desired circle. Then solving for x in mx = sqrt( (x-a)^2 + (mx-b)^2 ) gives x = a+mb - sqrt((a+mb)^2-(a^2+b^2)). FG is mb, AH is a+mb, and DI is sqrt((a+mb)^2-(a^2+b^2)).
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