TITLE: Is there any trilateral figure in euclidean geometry that is not a triangle?
QUESTION [7 upvotes]: I was going through Euclid's Elements and when i read definition 19 which says:

Rectilinear figures are those (figures) contained by straight-lines: trilateral figures being those contained by three straight-lines, quadrilateral by four, and multi-lateral by more than four.

and then definition 20 which says:

And of the trilateral figures: an equilateral triangle is that having three equal sides, an isosceles (triangle) that having only two equal sides, and a scalene (triangle) that having three unequal sides.

I couldn't help but wonder that why did Euclid use the term "Trilateral Figures" when he could have simply used the term "Triangles" and are there any "Trilateral Figures" other than triangles in Euclidean Geometry.

REPLY [9 votes]: Short answer: yes, here a trilateral means a triangle, though the reason of this distinction is not entirely clear. In other parts of the book, however, the word trigônon is used: for instance in proposition 47 (Pythagorean theorem).
It's interesting to have a look at Heath notes on this translation, as well as the original of the Heiberg manuscript (which is here, at the link you give in your question):

Heath writes (p. 187 of the first volume of his translation):

The latter part of this definition, distinguishing three-sided,
four-sided and many-sided figures, is probably due to Euclid himself,
since the words tripleuron, tetrapleuron, polypleuron do not
appear in Plato or Aristotle (only in one passage of the Mechanics and
of the Problems respectively does even tetrapleuron, quadrilateral,
occur). By his use of tetrapleuron, quadrilateral, Euclid seems
practically to have put an end to any ambiguity in the use by
mathematicians of the word tetragônon, literally "four-angled
(figure)", and to have got it restricted to the square.

I don't copy here the note on definition 20, but it's even more interesting, as it explains why, according to Proclus, there is a distinction between triangle and trilateral. Apparently it's due to a widespread misconception at that time, about the quadrilateral with one reentrant angle, incorrectly considered having only three angles.
As I understand it from Heath notes, there are thus several possibilities to explain this wording:

Have a similar word for $3$, $4$ or many sides.
Avoid using terminology that could be considered ambiguous due to an obscure paradox.
The most important property of a polygon is (considered to be) the number of sides, hence avoid a definition that is implicitly based on angles.

That said, it doesn't matter that you call this a triangle (three angles) or a trilateral (three sides): it's the same thing. Note that definitions 21 and 22 clarify the definition of a trilateral by naming the different cases of triangles.
The three volume translation by Thomas Heath can be found on Internet Archive, here.
While it's apparently a copy of the Dover books (same cover), these works are now in the public domain, so it's not a problem sharing the link.
It might be worth mentioning that Peyrard, in his french translation, uses the term figure trilatère, instead of the more common triangle. However, the order of definitions is slightly different: Heath's definition $19$ corresponds to Peyrard's definitions $20-23$. See here.