Ruler and Compass Construction 9
Dividing a Segment into Three Equal Parts

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1. Draw arbitrary slanted segment AC
2. Circle centered at A, arbitrary radius
3. Circle centered at D, radius AD
4. Circle centered at E, radius AD
5. Segment FB
6. Circle centered at F, through E
7. Circle centered at D, radius EG
8. Ray EH intersects AB at I
9. Circle centered at A, radius EG
10. Ray DJ intersects AB at K.
11. AK = KI = IB

Why it works:
By copying the angle EFB to D and E, lines DJ and EH are constructed parallel to FB. Thus
AF and AB are transversals to parallel lines, and so cut in equal ratios.