None of the solutions above reference a side length. Nor can there be a single solution no matter what you do. It is clearly scale invariant.
There is a question— since people seem unsatisfied with some of the solutions presented— to find the “shortest”, or the “most elegant” proof, and that is something that absolutely can be searched for on a computer these days.
Pleonast and Chronos’s solutions both start with “assign an arbitrary side length 1”. Obviously the value doesn’t actually matter, but you need some value to plug into the law of sines, etc. It just falls out at the end. My question is, basically, is there an identity of some kind that doesn’t reference a side length that provides the extra constraint in this problem?
That’s just a convenience. You could equally say something like, “Let a denote the length of this side” and express all the other lengths as multiples of a. You’re really dealing with the ratios of the side lengths, which is what trig is all about, after all.
Having said that, I don’t see what the most elegant proof would be.
I get all that. My point is that many or most problems of this nature don’t need to reference side length in any way. They can be solved with a small set of identities like that the angles in a triangle add up to 180 or that opposite angles on crossing lines are the same.
At first glance, this problem appears to be of that nature. There are lots of opportunities to use the angle-only identities. But it turns out they aren’t enough.
Stranger
I’m probably out of my depth here and these are topics I haven’t thought rigorously about in decades …
Do you mean “rigid” in the conventional sense or is there some geometric special use of the term I’m missing?
Seems to me all problems of this general nature assume the figure is 2D planar and not elastic in any way. Otherwise the degrees of freedom just explode.
I certainly understand your larger objection that it seems underspecified assuming you’re not going to dig into actual trig functions to resolve it.
I mean, in principle, you can avoid referencing side lengths. Every ratio of side lengths @Pleonast and I used could be replaced with the appropriate trig expression. But that’s just the same thing with extra steps.
It does feel like there ought to be a simpler way to do it; I agree with that.
I meant rigid in the sense that all angles and side length ratios are determined. Imagine that you have lengths of infinitely extensible rod and then some clamps that hold the ends at specific angles (also, this is Euclidean space and we’re keeping everything planar), as well as some that connect rod ends but allow a variable angle.
If you build a triangle out of these, with two constrained angles, then the whole system becomes rigid outside of scale (you can still make each rod 2x as long, say).
But if you build a quadrilateral with opposite angles constrained to 90 degrees, then the whole thing is not rigid in the same way. The remaining angles can be changed, and the side lengths can be any ratio you want. It’s not just the same shape with different scales.
I think I was wrong, though. The problem here is that line segment that connects the vertices with the X/Y angles and the W/Z angles. You can give them the wrong value and you can’t prove that it’s the wrong answer with non-trig tools (it seems). But I’m not sure if that makes it non-rigid, though.
I think the question can just be rephrased as “can it be solved without trig?” And if not, why not? What makes this problem special vs. otherwise similar problems that can be solved without referencing lengths (length ratios or otherwise)?
I’d suggest the answer is they cherry-pit-picked the given angles to leave the problem underspecified in angle-space.
A more typical problem cherry-picks the givens to ensure there is a path to the solution wholly in angle-space, even if it’s obscure enough to keep clever folks digging for a bit before they find it.
Whether that underspecification was deliberate is a different & harder question.
Isn’t “angle-space” the same as “side-length ratios” like @Chronos says? To make it concrete, imagine constructing a circle circumscribing the given triangle and specify its diameter to be 1. Then the length of each side in those units is simply equal to the sine of the opposite angle. Furthermore, “angle space” is the same as “sines-of-angles space” via right triangles as pointed out by @Stranger_On_A_Train .
So nothing is underspecified. It remains to figure out the shortest/best proof, though. I am actually curious about running this one through a computer.
Or, more fundamentally, that is expressed by similar triangles having the lengths of their sides in proportion.
Oh yes. The trig functions are operators to convert between “angle-space” and “side ratio space”.
And yes, the problem isn’t truly underspecified in the sense that there’s no way to compute the answer. It’s readily computable as has been shown upthread. Just introduce some “sideness” into the problem, take advantage of the trig operators, then the answer comes out and along the way the injected side info “cancels out” so there’s no loss of generality when you’ve gotten to the other end of the solution pipeline.
The “debate”, if there is one, is whether this problem should have an angles-only route to a solution. It doesn’t. But should it if it’s to fit in the class of traditional basic triangle geometry brain teasers?
That to me was what @Dr.Strangelove was getting at with his post just above mine.
And if not… why?
If you look again at the original diagram, every angle except for XYZW can be easily worked out. If you plug in an incorrect solution (that fits the angular constraints), how do you work out that it’s wrong? Say, X=50, Y=50, W=30, Z=80. You can show it’s wrong with trig. But seemingly not without it (I did try using @Stranger_On_A_Train 's suggestion of using the isosceles property, but I didn’t get anywhere).
Of course, it should be kept in mind that, whenever a human says that the Universe should be such-and-such a way, that’s actually a statement about us humans, and not actually a statement about the Universe.
Being able to solve it without trig seems to imply that the ratios of sides can be changed, keeping other knowns constant, without changing the answer. However that seems to necessarily change angles. So I vote no for answers without trig.
Not really. For a much simpler problem, consider a triangle with two angles given, and the third angle to be determined. The ratios of sides are fixed there, too, but one never needs to use trig to solve it.
If (hypothetically, but think of classical problems like trisecting the angle) the problem does not have a certain kind of solution, that should be proof-theoretically formalizable and proven or disproven. I really think the solution space (number of auxiliary constructions needed) in this case is small enough that a computer can brute-force it.
I was using should in the sense of the gamesmanship of folks who write problems for textbooks, quizzes, and internet brain teasers. This one seems inordinately hard which makes me wonder whether the test-writer goofed and intended XYZW to be soluble using angle-only reasoning.
Or, as I suggested upthread whether somewhere along the chain of plagiarizing from the original, some key tidbit of auxiliary info got lost or garbled along the way. So this goofed up revised version is unintentionally insoluble using only angles, whereas the original was soluble, and intentionally so.
You’re spot-on that the Universe cares not a fig for our puny human shoulds.
I now think @Dr.Strangelove is suggesting a deeper sort of “Why?” question than I was, namely: “What’s pathological about this particular collection of facts that prevents an angles-only solution?” What can we learn or say about classes of problems that do, or do not, admit of an angles-only solution? And other than brute force, how would we recognize which class any given problem belongs to?