Given three segments whose lengths are 1, a, and b, construct segments of length a + b, |a – b|, ab, , and . These are Euclidean constructions.
Hello, ReneePatt!
Surely, you can do the first two . . .
Given three segments whose lengths are:
construct segments of lengths:
On a horizontal line, measure off: .
Code:o - - - o - - - o - - - O 1 P a Q
Through draw a diagonal line.
On the diagonal, measure off
Draw
Code:* * * R * o b * / * / * / o - - - o - - - o - - - O 1 P a Q
Through construct a line parallel to , cutting the diagonal at
Code:S o x * / * / R * / o / b * / / * / / * / / o - - - o - - - o - - - O 1 P a Q
Then: .
Proof
From similar triangles: .
And we have: .
1. Draw a line with the length (a + b)
2. This line is the diameter of a circle.
3. Construct a perpendicular line at the end of a = begin of b. This line intersect the circle line. Construct a right triangle with (a + b) as hypotenuse .
4. According to Euclid's theorem you have in a right triangle:
Hello, Renee!
On a horizontal line, measure off: .
Code:o - - - o - - - o - - - O b P 1 Q
Through draw a diagonal line.
On the diagonal, measure off
Draw
Code:* * * R * o a * / * / * / o - - - o - - - o - - - O b P 1 Q
Through construct a line parallel to , cutting the diagonal at
Code:S o x * / * / R * / o / a * / / * / / * / / o - - - o - - - o - - - O b P 1 Q
Then: .
Proof
From similar triangles: .
And we have: .