Based on the following shape, and given AB, AC and BC are all known values, what is it possible to find out about the rectangle? Can I somehow deduce the sides and/or the area?
I'm working on a personal project involving some triangulation, so this is not homework :) Hope someone can shed some light!
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$A,B,C$ are fixed. Take a random straight line through C, not intersecting the segment AB. Take a perpendicular on that line, also through C. Take a parallel to the first line through A, and a parallel to the second line through B. As you can see, there are as many different rectangles as there are lines through C.
Here is a way to visualize the fact that we have infinitely many solutions, as Blue says in his comment: Fix the vertex C at the origin in $\mathbb{R}^2$. Put vertex $B$ at the point $(x,y)=(0,-|BC|)$, and use standard trigonometry to place $A$ in its place (I won't bother doing the calculations, but there is a unique place for it). This yields one possible rectangle. However, we can rotate the triangle about the origin, and so create many more, different rectangles, depending on how much we rotate. Since the set of all possible angles form a continuum, there are uncountably infinitely many rectangles achievable.
A quick sketch made in Geogebra to illustrate: