Little explorations with HP calculators (no Prime)

[Han]

(04-08-2017 09:17 PM)Gerson W. Barbosa Wrote:  I still don't understand all the fuss about the use of these particular shapes, since they would have spared lots of tedious calculations as we've seen :-)

For me, it had to do with the fact that without the most recent part being included as part of the solution, the special case is just that -- a special case. You may very well have done the general calculations for your own, and hence for you the special case was a shorter way to show the calculations. Mathematical proofs will sometimes leave out details and simply assert that a certain property is true. Usually, though, such assertions tend to be very simple to deduce. Granted, "simple" is very relative (a certain property may be obvious to one person with a lot of mathematical background; yet to others it may be difficult to see why the property is true). Nevertheless, "simple" is generally understood as "can be readily deduced after making a few quick observations that are either explicitly stated in the solution/proof or in the problem itself." However, when a solution only presents a special case without presenting how the general case can be reduced to the special case, many would consider such special cases to be an incomplete solution -- even more so when the seemingly missing pieces require a series of steps that are not obvious. In this case, one could reasonably argue that the tediousness of the algebra in using Heron's formula and in the law of cosines makes the reduction of the general case to the specific case a non-obvious (i.e. not "simple") statement that requires clear exposition. (Not to mention that Heron's formula was not used in the original solution to the special cases.)

I think that (in your first solution) if you had at least mentioned that the relative ratio of the areas remains the same regardless of the shape by considering the law of cosines and using Heron's formula (and thereby leaving the details as "an exercise for the diligent reader"), then your original solution would very well have been complete. (That was why I was repeatedly asking how you decided the shape did not matter -- because I myself and perhaps several others did not see the connection of the law of cosines and Heron's formula. I did use the law of cosines, but mainly to get the corresponding sine values, and certainly not the same way you did.)