Proving Heron’s formula by complex number (2013)
mathgarage.wordpress.com
mathgarage.wordpress.com
However, a complete proof really should explain the diagram at the top to properly define x, y and z.
The idea, if I understand correctly, is to draw the altitudes of the three triangles AIB, BIC, CIA, and then show that the six resulting right triangles are congruent in pairs because they have one equal angle at A/B/C (the segments AI, BI, CI bisect the angles of the triangle ABC), and the hypotenuse in common. Not too complicated but even if a picture is worth a thousand words, it wouldn't hurt to have it written also in words.
A subsequent step would prove that r is equal for all pairs of triangles, because it is a common side of triangles coming from different pairs. (As a bonus this proves that the incenter is indeed the center of the inscribed circle).
Likewise, going from r+ix to u*e^iα is basically Pythagoras's theorem, but it would not hurt to write that explicitly. And it does not hurt to write "let s=(a+b+c)/2=x+y+z be the semiperimeter of the triangle" (x+y+z is not the definition of the semiperimeter!) and to only introduce it after x+y+z actually occurs somewhere.
It's a really cool proof, and hey it was written by a high school student—but the proof writing is as important as the proof IMO.
My original writeup was longer and included several of the details that you mentioned. However, the editors of the American Mathematical Monthly condensed it, presumably so that it could be published as a "filler" that would take up less than a page. What you see is their rewrite of my work.
In fairness to them, the stuff that's left out is well known to people who think about incircles and excircles and thus not really new. For instance, it's used in the proof of Heron's formula by similar triangles.
There were some things that I would not have put that way on my own. For instance, at that point in my life, I had actually been trained not to use the word "clearly" like it's used here.
But who was I, a high school student, to quibble with the editors of the American Mathematical Monthly? I was thrilled that they wanted to publish my work. No regrets.
Make it a rant about the editors—if they had been your "mistakes" they should have corrected you, and I am happy to learn that they weren't yours in the first place.
Original pdf seems to have been taking down from AOPS but is still available in the internet archive: https://web.archive.org/web/20121018044426/https://artofprob...
https://www.tandfonline.com/doi/pdf/10.4169/amer.math.monthl...
But this time not with complex numbers.
It may not have a financial cost, but it certainly has an opportunity cost, and contributes to "Account Fatigue".
But loved the proof of Heron's formula, and I'm pleased to be able to say thank you in person, and make sure I give credit when I use it for the Math Club talks I do.
Great work ... feel free to contact me if you are so inclined, or put an email address in your profile.