I have two convex quadrilaterals (ABCD and WXYZ). Their sides and their interior angles are known. I also know that WXYZ fits perfectly inside ABCD with each corner point touching a different side.
Is there any way to figure out analytically where the touches occur?
For example, if I knew AX or angle BXY, everything else would be obvious because it is all triangles. But how to find out where X is on line AB?
... Of course, I can always draw the things on different pieces of paper, cut WXYZ out, fit it by hand and then measure the result. But that isn't terribly precise. (!!)
Assume a solution.
Let a = m angle A; b= m angle B; and so on.
Let $\theta = $ m angle AXW.
By law of sins:
$\sin a/WX = \sin \theta/AW = \sin(180 - \theta - a)/AX$
$\sin b/XY = \sin(180 - \theta - x)/BY = \sin(\theta + x - b)/XB$
$\sin c/YZ = \sin(180-y - \theta - x + b)/ZC = \sin(y + \theta + x - b - c)/YC$
and
$\sin d/WZ = \sin (\theta + a - w)/DZ = \sin (180 - d - \theta - a +w)/DW$.
(And we can also note: $180 - d -\theta - a + w = 180 - z - y - \theta -x + b + c$)
Using these was can solve AW,AX,BY,XB,ZC,YC,DZ, DW.