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This proof cannot be found in cut-the-knot.org nor in Loomis' collection.

Let $\triangle{ABC}$ be a right-triangle with $\angle{ACB}=90^\circ$. Let $D$, $E$ and $F$ be the contact points of the incircle with $BC$, $AC$ and $AB$, respectively. Also, let $AE=AF=x$; $BD=BF=y$; $CD=CE=r$, where $r$ is the inradius of $\triangle{ABC}$.

Assume to the contrary that $a^2+b^2>c^2$. Then,

$$(r+y)^2+(x+r)^2>(x+y)^2.$$

Expanding, collecting like terms and simplifying we get

$$ry+rx+r^2>xy.$$

Notice that $ry+rx+r^2=r(y+x+r)=rs=\Delta$, where $\Delta$ denotes the area of $\triangle{ABC}$ and $s$ its semiperimeter. Moreover, it is well-known that $xy=\Delta$. So $ry+rx+r^2>xy$ is equivalent to write $\Delta>\Delta$, which is a contradiction. A similar situation arise if you assume $a^2+b^2<c^2$.

I got this proof rejected by an editor because, accoding to him, this is the same proof as his, just more complicated since I presented it as a contradiction.

This is the editor's proof:

For any triangle, $\Delta=rs$. (Standard argument by areas)

So we have $\frac{1}{2}ab=(s-c)s=\frac{1}{4}(a+b-c)(a+b+c)$ and this, after a couple of lines of simple algebra, yields $c^2=a^2+b^2$.

Is this really the same argument? Is always proof by contradiction more complicated than direct proofs?

Emmanuel José García
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2 Answers2

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"Is always proof by contradiction more complicated than direct proofs?" Always? No. Here? Yes.

Your assumption that $a^2+b^2>c^2$ has no bearing on how the proof unfolds. You seem to assume an inequality just to have two sides of an in/equation to manipulate in tandem, which is unnecessary.

Without making any assumption, you could walk through your proof replacing "$>$" with, say, "$\bigcirc$" to represent an unknown, and unassumed, comparator ($>$, $<$, or $=$). So, you start with $a^2+b^2\bigcirc c^2$, and end with $\Delta\bigcirc\Delta$. This reveals that "$\bigcirc$" must have been "$=$" all along, and you're done. No contradiction required.

Of course, the "$\bigcirc$" formulation is a little unusual; all it really affords us is that two-sided in/equation structure, but we can do without that. A better approach is to combine those sides, and to consider the nature of the expression $a^2+b^2-c^2$. Unpacking, we get

$$a^2+b^2-c^2=(r+x)^2+(y+r)^2-(x+y)^2=2\left(\;r(y+x+r)-xy\;\right)=2\left(\;\Delta-\Delta\;\right)=0$$

This gives the result, again without contradiction.

Can you see how your contradiction is somewhat "artificial"?

For further opinions about the relative (de)merits of proofs-by-contradiction, search the site. You'll find questions like these

Blue
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  • I posted my comment here https://geometriadominicana.blogspot.com/2020/05/my-response-to-blue-and-david-k.html since it does not fit here. – Emmanuel José García May 15 '20 at 14:50
  • Do you think the introduction of the comparators <, >, make my proof more complicated because it is more natural to use =? You complain about the presence of the contradiction but do not explain HOW it makes my proof less simple. You say the comparator is not necessary, but that does not make any sense to someone who learn the Δ=xy and is capable of using it in other contexts by going my way. Wait a minute, is not that informative aspect that make pro-direct-proof mathematicians prefer direct proofs? Hmmm... – Emmanuel José García May 17 '20 at 01:10
  • "Of course, the "◯" formulation is a little unusual" That's not a little unusual in elementary math, that's extraterrestial. – Emmanuel José García May 17 '20 at 01:13
  • @EmmanuelJoséGarcía: My point with the "$\bigcirc$" formulation is simply that you needn't assume the comparator; you can deduce it. Your contradiction argument, while valid, adds an unnecessary wrapper. (As for the "extraterrestrial" notation: I recall from the most-elementary of math classes seeing the likes of "$7;\bigcirc;5$" and having to fill-in the appropriate comparator. So, I consider my notation "nostalgic". ;) ... Take or leave my opinion as you will. I'm disinclined to argue with you, especially in this cramped comment space. I won't be responding further. Cheers! – Blue May 17 '20 at 02:08
  • Diamonds are unnecessary but beautiful. Cheers! – Emmanuel José García May 17 '20 at 02:25
  • Btw, the same critics can be apply to proofs sixteen and thirty-two in Loomis's collection (both in a section called "Noted proofs") and to Subramaniam's proof in Math. Gaz. 102 (2018). All published proofs by contradiction! – Emmanuel José García May 17 '20 at 02:50
  • No one is saying that proofs by contradiction aren't publishable. Even so, if an argument looks like "(1) assume $x\neq 5$, (2) proof that $x=5$, (3) contradiction!, hence $x=5$", then clearly the contradiction aspect only adds cognitive clutter that should be avoided. Generally, it's worth asking whether one's own contradiction has such clutter. With Loomis #16/2 —which assumes, for one right triangle, that "$hyp^2\color{red}{<}leg^2+leg^2$", and deduces, for a similar triangle, that "$hyp^2\color{red}{>}leg^2+leg^2$"— the contradiction comes across as clever and striking, not clutter-y. – Blue May 17 '20 at 05:02
  • No many people are familiarized with the Δ=xy. My proof can lead you to encounter with this expression and ask yourself, what is that situation in which $rs>xy$ is a contradiction? You may consider the posibility that $\Delta=xy$ and work to prove it true. In this sense, my proof is as potentially informative as the direct proof, which is a good point if you consider this is the main reason why some mathematician prefer direct proofs over indirect proofs. On the other hand, the ability to find contradictions can be extrapolated to other contexts of daily life: debates, for example. – Emmanuel José García May 17 '20 at 14:08
  • I wonder if pro-direct-proof mathematicians would also prefer a more complicated proof, but more informative, over a simpler, but less informative, proof. – Emmanuel José García May 17 '20 at 14:08
  • Your argument about cognitive clutter seems to me very subjective. Some people prefer number theory over geometry because the latter seems to them very confusing/difficult, and vice versa. Some mathematicians would not agree with you about Loomis #16/2. – Emmanuel José García May 17 '20 at 14:33
  • Also, I have not seen you apply any criteria of simplicity to both proofs. For example, the Law of Parsimony fails to show that one is simpler than the other because both proofs start with two propositions: $\Delta=rs$ and $\Delta=xy$ (for the direct proof); the two-tangent theorem and $a^2+b^2>c^2$ (for the indirect proof). Does the indirect proof requires more steps? No! Is, in this case, the indirect proof as potentially informative as the direct proof? Yes! This seems to be your believe: Ok, this is clear/easy/evident to me, therefore, it must be clear/easy/evident to everybody. – Emmanuel José García May 17 '20 at 15:07
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    @EmmanuelJoséGarcía: Yes, it's all very subjective, hence not worth extended debate. I'm done here (again! :). – Blue May 17 '20 at 15:14
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Here's a direct proof using your symbols and formulas.

Let $\triangle{ABC}$ be a right-triangle with $\angle{ACB}=90^\circ$. Let $D$, $E$ and $F$ be the contact points of the incircle with $BC$, $AC$ and $AB$, respectively. Also, let $AE=AF=x$; $BD=BF=y$; $CD=CE=r$, where $r$ is the inradius of $\triangle{ABC}$. Let $\Delta$ denote the area of $\triangle{ABC}$ and $s$ its semiperimeter.

Then $ry+rx+r^2=r(y+x+r)=rs=\Delta$ and it is well-known that $\Delta = xy,$ so $$ ry+rx+r^2 = xy.$$ By algebraic manipulation (on each side multiply by $2,$ then add $x^2 + y^2,$ then factor) we obtain $$(r+y)^2+(x+r)^2 = (x+y)^2,$$ that is, $a^2 + b^2 = c^2.$

Sometimes direct proof is difficult and proof by contradiction is simpler. In this case it is not.

You might have had more luck presenting a direct proof. I would argue that your proof is different because you require knowing not only that $\Delta = rs$ but also $\Delta = xy.$ This is not necessarily a point in favor of your proof; but my impression of the sites that have large collections of proofs is that they look for variation in methods, not strictly for optimality.

David K
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  • I posted my comment here https://geometriadominicana.blogspot.com/2020/05/my-response-to-blue-and-david-k.html since it does not fit here. – Emmanuel José García May 15 '20 at 14:52
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    Presumably your complaint about notation does not apply here. This proof has the exact same notation and all its steps come from your proof, although some are in the opposite direction since this proof is direct. You also take offense at my comment about needing to know $\Delta=xy.$ Indeed that is merely an opinion. You may feel you benefited from trying this way. But it is the editor of the proof site whom you must convince of your proof's merits, not me. – David K May 15 '20 at 20:44
  • Anyway I answered “no” to your question, “Is this really the same argument?” Was that not the answer you wanted to that part of your question, at least? – David K May 15 '20 at 20:50
  • You seem to believe that the presence of ">" is not natural and conclude my proof could not be simpler, as if all natural are always simpler than non-natural. Suppose that all natural are always simpler than non-natural, then, walking 10 km for pizza is simpler/easier than taking the bus, which is absurd. On the other hand, we start both with the same number of facts: Δ=rs and Δ=xy (in your case) and two-tangent theorem and a^2+b^2>c^2 (in my case). Both proofs have the same number of steps and mine can be as potentially informative as yours. So we cannot say which of this proof is better. – Emmanuel José García May 16 '20 at 12:48
  • What does "natural" have to do with anything? Anyway, what I wrote is merely what your proof might be without using contradiction. And what's all this about walking 10 km for pizza? When we compare your proof (or my version of it) to the editor's proof, did the editor really work harder? – David K May 16 '20 at 13:23
  • The point is you did not explain why yours is simpler. – Emmanuel José García May 16 '20 at 13:29
  • What I wrote is deliberately made almost identical to your proof. If you like the bus analogy, your first argument is that if you get off the bus too late you are past the pizza shop, and if you get off the bus too early you have not yet reached the pizza shop, whereas I just took the bus and got off at the right stop. – David K May 16 '20 at 13:36
  • What does "natural" have to do with anything? This is me trying to figure out what are the reasons make you think your proof is simpler. Btw, you edited your answer to add-among other things-the "not necessarily" part, which in others words means that "this could be a point in favor of my proof". Thanks for admitting it. Still waiting you explaing why my proof is more complicated. – Emmanuel José García May 17 '20 at 00:53
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    We are getting into matters of opinion and there is no final resolution to those. Sometimes it is a benefit to use an advanced piece of knowledge because it shortens the proof. But if it does not actually reduce the work required to complete the proof, what is the benefit? On the other hand, who decides whether the work is actually reduced? And what does that have to do with whether a proof is worth including in a collection such as the one you submitted it to? – David K May 17 '20 at 12:15
  • "what is the benefit?" Well, as I told you, no many people are familiarized with the Δ=xy. My proof can lead you to encounter with this expression and ask yourself, what is that situation in which $rs>xy$ is a contradiction? You may consider the posibility that $\Delta=xy$ and work to prove it true. In this sense, my proof is as potentially informative as a direct proof, which is a good point if you consider this is the main reason why some mathematician prefer direct proofs over indirect proofs. – Emmanuel José García May 17 '20 at 13:50
  • On the other hand, the ability to find contradictions can be extrapolate to other contexts of daily life: debates, for example. – Emmanuel José García May 17 '20 at 13:51