Let ABCD be a cyclic quadrilateral with side lengths a, b, c, and d.

What is the area ( in terms of a, b, c, and d ) of the convex hull of
four circles ( with centers A, B, C, and D and each passing through
the circumcenter of ABCD )?
  

The figure can be broken down into 9 separate pieces:
The quadrilateral itself (call this area X),
Sectors of the four circles with radius equal to that of the circumcircle of ABCD (call this radius r),
and four rectangles with areas ar, br, cr, dr.

The four sectors together make a complete circle.

Without finding X and r we can express the area as

X + pi*rē+ r(a+b+c+d)

The formulas of X and r can be found at http://mathworld.wolfram.com/CyclicQuadrilateral.html (4) and (8)

Posted by Jer on 2014-09-11 11:56:14