Ruler and Compass Construction 14
Inscribed Circle of a Given Circle

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1. Bisect angle A
2. Bisect angle B
3. Intersection O, center of the incircle
4. Ray OM perpendicular to AB (construction 2)
5. Circle centered at O, through N, is the incircle

Why it works:
The center of the incircle must be on the angle bisector of each angle, so the construction
finds two angle bisectors. Then to find the radius, it does the perpendicular from the center O to the side AB.