Proposition VI.10 — To cut a given uncut straight line similarly to a given cut …

Proof

Lay the cut transversal alongside the given line meeting at one endpoint; join the far ends and draw parallels to that join through each cut point (I.31). By VI.2 the parallels reproduce the same ratios on the given line.

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Depends on (2)

I.31Proposition I.31Through a given point to draw a straight line parallel to a given straight line.
VI.2Proposition VI.2If a straight line be drawn parallel to one of the sides of a triangle, it will cut the sides of the triangle…

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