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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).$

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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 $$ a\sin x +b\sin y = \frac{2S}{c}. $$

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  • $\begingroup$ OK. I deleted it. $\endgroup$
    – Daniel N
    Nov 24 '20 at 17:39

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