Question

Assume that you are given segment AB of length 1 (a unit). Given line segment of length a , b and c construct a line segment of length abc.

Answer 1

Assuming 1<a<b<c.

1. Construct a triangle ABC right angled at B such that AB = a and BC = 1.
   
2. Extend CB to D such that CD = b
   
3. Construct DX perpendicular to CD.
   
4. Extend CA such that it cuts DX ate E.
   
   DE = ab

Similarly, we contruct length abc from length ab using the same steps. Just replace a with ab and b with c.

1. Construct a triangle A'B'C' right angled at B' such that A'B' = ab and B'C' = 1.
2. Extend C'B' to D' such that C'D' = c
3. Construct D'X' perpendicular to C'D'.
4. Extend C'A' such that it cuts D'X' ate E'.
   D'E' = abc