4 ms·
I'm okay with conjuring (r+ix)(r+iy)(r+iz) out of thin air. I am also okay with adding an extra step at the bottom to make it a proof of Heron's formula, even t
by bonzini 3y ago
I'm okay with conjuring (r+ix)(r+iy)(r+iz) out of thin air. I am also okay with adding an extra step at the bottom to make it a proof of Heron's formula, even though at its heart it's a proof of the radius of the incircle—because the former sounds way more cool.
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.
- mehulashah 3y agoInteresting. Certainly, those small leaps are missing. But, that’s what made the proof fun for me. They were small enough that I could fill in the blanks. In some cases, it took a little thinking, but overall it was a nice way of approaching the construction.
- milesde 3y agoHi! I wrote that proof back in 2007, half of my lifetime ago. It's neat to see it discussed again. 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.
- bonzini 3y agoHey it's nice that you found this post and replied! You shouldn't have any regrets indeed, and I guess that style has its place for a "filler" article, even though calling it a filler is not so nice... 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.