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
$\endgroup$ 52 Answers
$\begingroup$A CW answer to remove this question from the Unanswered queue.
Yes, "triangle" can be taken to mean "non-degenerate triangle" unless explicitly stated otherwise. As far as I know, this convention is universal.
$\endgroup$ 0 $\begingroup$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 and 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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