An Incorrect Proof of an Impossible Triangle
Why an old email I got teaches us how to think more clearly (even though it was wrong)
Maybe a year ago, I got an email from an older guy I know, who complained about his daughter’s online math homework. He was convinced that the learning software was wrong, and he was right. I’ve known a few parents over the years who complained about their children’s online software being wrong, and I can’t remember a single time where the parent was right. But still, I wanted to listen.
His daughter was in middle school, and the question was straightforward:
Can you create a triangle with sides of lengths 6, 7, and 13?
His daughter’s answer was yes, and the program’s answer was no. The dad ranted to me about how wrong the software was, even proclaiming that this was “math written by Literature majors.” That was kind of weird. But then the dad gave me a “proof” he wrote of why 6,7, and 13 can form a triangle. This was all I remember from the “proof”:
“The Law of Cosines is a generalization of the Pythagorean Theorem which says that in a triangle, the hypotenuse is equal to c²=a²+b²-2ab(cosθ). Blah blah lengths, blah blah take these sides and multiply blah blah… meaning 6,7, and 13 form a triangle. QED.”
There’s a good reason I don’t remember the proof after the first sentence: I didn’t have to read the rest of it. As soon as I saw “Law of Cosines,” it was clear that this proof was incorrect. And I’ll explain why.
The Triangle Inequality
Let’s just get this out of the way first: sides of lengths 6,7, and 13 can not form a triangle, because of something called the triangle inequality.
The triangle inequality basically says that any two sides of a triangle have to be larger than the third side. In other words, if the lengths of the sides are a, b, and c, then these have to be true for the sides to make a triangle:
a<b+c
b<a+c
c<a+b
You will sometimes see these formulas with a ≤ sign instead of a < sign (for example b≤a+c), but in the case where b=a+c, the “triangle” isn’t really a normal triangle. If b=a+c, that “triangle” would look like a straight line:
So in K-12 geometry classes (and online math software), they usually teach the triangle inequality as a<b+c instead of a≤b+c, because we don’t naturally think of a straight line as a true triangle. From here it’s easy to see why 6,7, and 13 do not form a triangle: because 13<6+7 is not true. 13 is not less than 13. So the software was right. The sides cannot form a triangle. But now let’s get to the error in the “proof.”
Why The Proof is Wrong
Almost any math proof uses the idea of implication: this thing implies that thing. If some event P is true, then event Q is also true. This is often written as P→Q or “P implies Q.”
For example, event P could be “It’s raining.” Event Q could be “My lawn is wet.” Here, P implies Q: if it’s raining, then my lawn will get wet.
“P implies Q" means if P is true, then Q must be true. But if P is false, Q may or may not be true. Suppose it’s not raining out one day (P is false). The event Q could still be true, perhaps if I water my lawn with a hose that day. Or Q could be false if I don’t water the lawn that day. The point is that if P isn’t true, we can’t be certain one way or the other whether Q is true.
I’m going to rephrase the Law of Cosines a bit. See if you can spot the problem with using it to prove something is a triangle:
The Law of Cosines: If a, b, and c are the sides of a triangle, then c²=a²+b²-2ab(cosθ), where θ is the angle between sides a and b.
There’s an implication here! If a, b, and c form a triangle, then we know c²=a²+b²-2ab(cosθ). The dad used the Law of Cosines to “prove” that 6, 7, and 13 form a triangle. But the Law of Cosines only works if you already know you’re working with a triangle. And the entire homework question was about whether 6,7, and 13 form a triangle in the first place.
In other words, the dad did the proof backwards. He used c²=a²+b²-2ab(cosθ) to show that a, b, and c are a triangle, when the Law of Cosines works in the opposite direction:
One thing that often trips up math students is that just because P→Q, that does not automatically mean Q→P.
It might be true that if it’s raining (P), then my lawn will get wet (Q), or P implies Q. But if my lawn is wet, does that mean then it’s raining out? Does Q imply P? Not necessarily. I can make my lawn wet with a hose, or a sprinkler, or maybe an inflatable kiddie pool accidentally popped in my yard. There are many scenarios here where Q is true but P is false. It’s not necessarily true that Q implies P (Though some pairs of events genuinely do imply each other, and this is written as P↔Q or “if and only if P then Q”).
The Law of Cosines only applies to triangles; it’s useless if your 3 sides don’t form a triangle. From there, it doesn’t matter what else is in the proof. We know that the proof is wrong, because it’s based on the Law of Cosines, which can’t be applied if we’re not working with a triangle (and we’re not working with a triangle, which we know from the triangle inequality).
The Takeaway
If you use some mathematical rule, make sure it can actually be applied. The Law of Cosines only applies to triangles. As another example, the Pythagorean Theorem can only be applied to right triangles: if a,b,c are the sides of a right triangle (and c is the hypotenuse), then a²+b²=c². I can’t just take any three random numbers (my height, my weight, and my credit score) and say that a²+b²=c² must be true.
It’s easy to forget this, and I often make that mistake too. I explained this, more or less, to the dad in a reply. The mistake in the proof was that the Law of Cosines couldn’t be applied. I never found out if his daughter got the next homework question right, but I bet she did.
