2:3:4, 3:4:5, 4:5:6

[Ed Pegg]

To my great surprise, the 2:3:4 triangle is a sporadic similar triangle.  


The 3:4:5 triangle is a staple triangle known by all.  

The 4:5:6 triangle is one of only 12 triangles that can be divided into 5 similar triangles (though not similar to itself).  

[Fred Lunnon]

  Initial reaction:  2 3 4, 3 4 5, 4 5 6  --- yeah, beginning to see a pattern developing here ... 

  Wait a minute  ---  2:3:4  ??!!  That's barely credible! 

WFL 

_

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[Allan Wechsler]

I'm as startled as Fred. If we label the angles of this triangle a, b, c, with a the smallest and c the largest, is it really true that c = 2a + b?

There is a whole continuum of triangles with that angle relationship. Do they all admit this sporadic tiling?

-- Allan

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[Arthur O'Dwyer]

On Thu, Oct 8, 2026 at 9:31 AM Fred Lunnon <fred....@gmail.com> wrote:

  Initial reaction:  2 3 4, 3 4 5, 4 5 6  --- yeah, beginning to see a pattern developing here ... 

And don't forget that the 1:2:3 triangle can be divided in an infinite number of ways into any number of triangles similar to itself, all having the same area as itself. ;)

–Arthur

[Allan Wechsler]

Confession: I asked ChatGPT, and it claims, plausibly, that the 2:3:4 ratio of the sides does impose the relationship 2a + b = c on the angles. It's an elementary application of the law of sines which I am, at the moment, too groggy to confirm for myself.

Question: Other than (2,3,4), how common are such "Peggian" triples, where the large angle is an integer linear combination of the small ones?

Even if the sides aren't in a neat ratio, one certainly expects families of triangles satisfying such angle relationships to admit occasional "sporadic" dissections.

By the way, the frightening central intersection where it's required that 3c + b = 2pi -- it turns out that follows immediately from 2a + b = c. I wasn't too groggy to do that.

-- Allan

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[Geoffrey Caveney]

There is more information about the 2-3-4 triangle at this link:

https://math.stackexchange.com/questions/2696555/property-of-the-2-3-4-triangle-and-its-extensions

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