# How do sides of a constant area quadrilateral rotate? [closed]

A quadrilateral of constant area having three fixed length sides $$(a,c,b)$$ has fourth side length variable. End coordinates of fixed side length $$c$$ are $$(0,0),(c,0)$$.

Find relation between angles $$(x,y).$$

Let $$S$$ be the constant area of the quadrilateral. We consider the right trapezoid obtained by projecting the vertices of the variable edge on the horizontal line. We have then $$S = \frac{1}{2}[(a\sin x +b\sin y)(c -a\cos x +b\cos y) +a^2\sin x \cos x -b^2\sin y\cos y],$$ hence the relation $$a\sin x +b\sin y = \frac{2S}{c}.$$