Finding angle of parallelogram with circle passing through 3 vertices

Jan 2016
Given the above construction of parallelogram ABCD, with point E between B and C lying on the circle, the sine of angle A must be greater than √(x)/y where x and y are positive integers. What is the greatest possible value of √(x)/y? Answer: √(5)/3

At first I considered using the law of cosines and took the sine of the arccosine of (122 + 162 - BD2)/(2*12*16) on the domain [4, 28] because of the triangle inequality with 4 ≤ BD ≤ 28. I got a maximum value of sin(A) = 1 at BD = 20. However, it's obvious this is not fully correct since I imagine the parallel sets of lines and circle itself would offer some other upper constraint that I can't figure out. Even once I figured out the maximum of sin(A), how would I find the maximum value of √(x)/y with x and y as integers, or what might √(x)/y represent in terms of the problem given?