To find the area, but the easiest approach is using Brahmagupta’ s formula1:

Question

If a quadrilateral of side lengths a, b, c, d can be inscribed in a circle, then its area is given by
A = sqrt<(s − a)(s − b)(s − c)(s − d)>, where s =
(a + b + c + d)/2

In this case, a = 2*radius
You know b and d are equal, and you can write a relationship, I think between b and c (c= d - 2b*cosAngle) and you can figure the cosine of the angle in terms of the secants.

Then maximize the area, set to zero, and solve for b or c.

Could be some nasty math. I don't see an easier analytic method.

find the dimension of the trapezoid of greatest area that can be inscribed in a semicircle of radius r