Euclid's Elements II.11
Euclid's Elements, Book II, proposition 11:
To cut a given straight line so that the rectangle contained by the whole and one of the segments is equal to the square on the remaining segment.
Process:
0. Given line AB
1. Draw square ACDB on AB.
2. Bisect AC at E
3. Notice that EB is the hypotenuse of a right triangle.
4. Construct EF to be congruent to EB. F will be on the opposite side of A from E
5. Choose H on AB so that AF is congruent to AH.
Done. H is the point for which the rectangle, AB*BH = AH^2
The construction; drag A and B to see the square equals the area
Now think about this (try to solve before Thursday!)
Since AB is a given line segment, let's call it . The point we are constructing is going to be somewhere that we don't know at first, let's call it .
What is the algebraic expression that is suggested by the proposition? And what is the solution?