Ruler and Compass Construction 17
Divide a Segment AB in the Same Ratio as E Divides CD

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1. Arbitrary ray AF
2. Copy length CE to AG
3. Copy length CD to AH
4. Segment HB
5. Copy angle at H to G
6. Ray GL parallel to HB
7. AM/AB = CE/CD

Why it works:
The idea is to use the theorem that parallel lines cut transversals in equal ratios.
Thus line GM is constructed to be parallel to line HB, and one can trivially assume
a parallel line runs through A, so AB gets cut in the same ratio as AH.