# Does “triangle” in English exclude degenerate triangles?

Just for fun read few problems on the projeteuler.net website.

Number 276 found interesting:

Consider the triangles with integer sides a, b and c with a ≤ b ≤ c. An integer sided triangle (a,b,c) is called primitive if gcd(a,b,c)=1. How many primitive integer sided triangles exist with a perimeter not exceeding 10 000 000?

Question:
Does the definition "triangle" in English also enforces the following condition?

a + b > c

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Yes. If it didn't, then $a$, $b$ and $c$ could not form a triangle. –  Chris Taylor Mar 2 '12 at 16:23
It must hold that $a + b \geq c$. $a + b = c$ holds iff the triangle degenerates into a line (two points coincide). Usually, when one says triangle, one means a non-degenerate triangle. –  aelguindy Mar 2 '12 at 16:24
@aelguindy so in this particular problem can I assume that we are talking about non-degenerate triangles? –  Dime Mar 2 '12 at 16:27
@Dime: Absolutely you can assume non-degenerate. –  André Nicolas Mar 2 '12 at 16:39
Of course, the real place to ask what a Project Euler question means is on the Project Euler site, not here. –  GEdgar Mar 2 '12 at 18:29

Yes, "triangle" can be taken to mean "non-degenerate triangle" unless explicitly stated otherwise. As far as I know, this convention is universal.

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In some mathematical English, a triangle means exactly what it did 2,300 years ago:

"A triangle is plane figure bounded by three straight lines." (A Text-book of Euclid's Elements, Hall & Stevens, 1888).

This definition seems unclear.

But later, Proposition 20 clarifies it (emphasis added):

"Any two sides of a triangle are together greater than the other third side." (ibid.)

However, the articles https://en.wikipedia.org/wiki/Triangle_inequality and https://en.wikipedia.org/wiki/Normed_vector_space contradict one another: the latter includes equality.

But, I think, most (English) mathematicians would agree with the latter, and disagree with Euclid, and will allow degenerate triangles to partake in the so-called triangle inequality.

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