Solve Abu Kamil's Algebra Puzzle from 9th/10th Century

[AKG]

"One says that ten is divided into three parts, and if the small one is multiplied by itself and added to the middle one multiplied by itself, it equals the large one multiplied by itself, and when the small is multiplied by the large, it equals the middle multiplied by itself."

This is a problem of 9th/10th century Islamic mathematician Abu Kamil. I found it in a an issue of Science in a review of "Unknown Quantity", a book about the history of algebra. It kept me entertained while I was waiting for a professor, see if you can figure it out. On a side note, the author's response to the review of his book, along with a reproduction of the review itself, are here.

 

[0rthodontist]

a <= b <= c
a + b + c = 10
aa + bb = cc
ac = bb
aa + ac = cc
a/c + 1 = c/a
let x = a/c
x + 1 = 1/x
xx + x - 1 = 0
x = (-1 + sqrt(1 + 4))/2 = a/c
xcc = bb
b = sqrt(x)c
cx + sqrt(x)c + c = 10
c = 10/(x + sqrt(x) + 1)
a = cx
b = sqrt(x)c

 

[tacman]

After examining the puzzle, it seems that we are being asked to find three numbers that satisfy the given conditions. Let's start by assigning variables to the three parts mentioned in the puzzle. We can call the small part "x," the middle part "y," and the large part "z."

Based on the given information, we can create the following equations:

x + y + z = 10 (since the parts add up to 10)
x^2 + y^2 = z^2 (since the sum of the squares of the small and middle parts equals the square of the large part)
x * z = y^2 (since the product of the small and large parts equals the square of the middle part)

Now, we have a system of three equations with three variables. We can use algebraic techniques to solve for the values of x, y, and z.

First, let's rearrange the second equation to isolate z^2:

z^2 = x^2 + y^2

Then, substitute this expression for z^2 into the third equation:

x * z = y^2
x * (x^2 + y^2) = y^2
x^3 + xy^2 = y^2

Next, rearrange the first equation to isolate z:

z = 10 - x - y

Substitute this expression for z into the second equation:

x^2 + y^2 = (10 - x - y)^2
x^2 + y^2 = 100 - 20x - 20y + x^2 + 2xy + y^2
0 = 100 - 20x - 20y + 2xy

Finally, we can solve for y in terms of x by rearranging this equation:

20y = 100 - 20x + 2xy
y = (100 - 20x + 2xy)/20
y = 5 - x + (xy)/10

Now, we can substitute this expression for y into the first equation:

x + (5 - x + (xy)/10) + (10 - x - (xy)/10) = 10
5 - x + (xy)/10 = 0

Solving for x, we get x = 5. Then, substituting this value into the expression for y, we get y = 5. Finally,