Talk:Equilateral triangle

Untitled
'''IS AN EQUILATERAL TRIANGLE CALLED AS A PERFECT TRIANGLE ? IF YES, WHY ?'''
 * The term perfect triangle seems to have a few different definitions see for example which defined perfect triangle as triangles with sides of integer length and having numerically equal integer area and perimeter. An equlatrial triangle would never satisfy this definition. --Salix alba (talk) 14:25, 13 April 2007 (UTC)

Area of Equilateral Triangle
I do not know too much about geometry, so I will post a suggestion for a change. The article states that the area of an equilateral triangle is 1 $${\sqrt{3}\over 4}$$, where 1 is the lenght of a side. I understood it to be s2 $${\sqrt{3}\over 4}$$,where s = side. Of course the given example is 1, so having it squared will make no difference. But given any other number, and it will need to be squared. Is this not a better way of writing the equation? --Mateck 01:33, 2 May 2007 (UTC)

They said it was a duplicate of a formula. --Milesman34 —Preceding undated comment added 13:07, 11 November 2016 (UTC)

Equilateral Triangles cannot have all integer planar coordinates (not embeddable into Z^2)
Please consider adding this fact to an appropriate section of the article.

Equilateral triangles cannot be formed from the grid points of a regular two-dimensional lattice, such as on graph paper, or in software with integer (x,y) coordinates.

This is related to the irrationality of sqrt(3). A number of proofs exist online, and there's this journal article which could serve as a trusted primary source:

Triangles with Vertices on Lattice Points Michael J. Beeson, The American Mathematical Monthly, Vol. 99, No. 3 (Mar., 1992), pp. 243-252

I suspect, though, that the result on Z^2 is much older.

A nice project for someone who wants to do some researching and editing... — Preceding unsigned comment added by 66.85.230.203 (talk) 02:56, 14 August 2022 (UTC)

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About characterizations
If you think that these are the same as the properties, then you should not be editing here, but instead learning what a characterization is! Properties are the necessary conditions that are true in an object, whereas characterizations are both necessary and sufficient conditions. The difference is that knowing that one of the properties of an equilateral triangle hold in a general triangle does not say that it is equilateral, but a characterization does say so. Thus a characterization is a unique property that an object and no other object has. It is logical to have a list of fundamental properties first, but that can never replace the list of characterizations. Should the latter be among the first sections or at the bottom is a matter of taste, but please do not remove this section again due to the ignorance that it is superfluous. Circlesareround (talk) 00:24, 17 April 2018 (UTC)