Best to start with a picture.A circle radius = 725 contains 4 isosceles
trapezoids, length of shorter parallel sides = 666.
Heights, other parallel sides, and equal sides are all integers.
What are the 4 heights?
Yes, you are right. My mistake was that I dropped a factor of 2 in the equation $725^2-644x-333y = \Box.$ It should have read $2(725^2-644x-333y) = \Box.$ I then get the values of $x$ to be $\pm364$ and $\pm500$, giving the heights as 144, 280, 1008, 1144.Nope...with your 4, the 2 equal sides AD and BC are not integers.
These are my 4 (height, AD/BC, CD, x):
144, 240, 1050, -500
1144, 1160, 1050, 500
280, 406, 1254, -364
1008, 1050, 1254, 364
Using your diagram:
AB = a, CD = b, height HK = h, OK = x, radius = r
I came up with this formula to derive:
r = SQRT(4x^2 + b^2) / 2 where x = (a^2 - b^2 + 4h^2) / (8h)
I don't know why you listed all the triangles you did;
we need triangle 333-644-725 for all cases;
then we need the triangles with a>333:
364-627-725, 435-580-725 and 500-525-725.
That's it, that's all: right?