** W------------------------X
| * |
| A * |
| O |
| * * * B |
| * * * |
| * * * |
| * * * |
| * D * C * |
Z----------------Q-------Y
**

What is the minimum area of rectangle WXYZ if

**all** lengths are whole numbers, as are the areas of the similar triangles, denoted by A, B, C & D?

Let a = QY, b = OQ, c = OY, d = QZ, e = OZ, f = OX, g = XY, h = WX, i = XZ

Then a series of equations can be written using only the Pythagorean theorem and similar triangle theorem:

c = sqrt[a^2+b^2]

d = b^2/a

e = sqrt[d^2+b^2]

e = b*c/a

f = c*a/b

g = sqrt[c^2+f^2]

h = a+d

h = g*b/a

h = sqrt[c^2+e^2]

i = e + f

i = sqrt[g^2+h^2]

The system of equations is redundant and simplifies to:

c = sqrt[a^2+b^2]

d = b^2/a

e = b*c/a

f = a*c/b

g = c^2/b

h = c^2/a

i = c^3/(a*b)

For any integral Pythagorean triple (a,b,c), the parameterization gives a rational solution. To ensure an integral solution, each value needs to be multiplied by a*b. This results in:

c = sqrt[a^2+b^2]

QY = a^2*b

OQ = a*b^2

OY = a*b*c

QZ = b^3

OZ = b^2*c

OX = a^2*c

XY = a*c^2

WX = b*c^2

XZ = c^3

Using a=3, b=4, c=5 gives QY=36, OQ=48, OY=60, QZ=64, OZ=80, OX=45, XY=75, WX=100, XZ=125. The area of the rectangle is 7500, the area triangle A is 3750, the area of triangle B is 1350, the area of triangle C is 864, the area of triangle D is 1536.