Pushing the limits

I’m still playing around with the ideas from going-one-step-beyond-herons-formula   And this time I looked at a more complex problem.  Overall the approach leaned heavily on the multiple area formulas this time (that lets Heron’s formula comes in).  But there was a fair amount of algebra in the middle so I’m not sure I like this method as much here.  The algebraic simplification may take as much logic as going for a more conventional approach.  The one upside is the conversion between various area formulas is quite interesting.