Talk:Equilateral triangle

"Perfect triangle"

IS AN EQUILATERAL TRIANGLE CALLED AS A PERFECT TRIANGLE ? IF YES , WHY ? — Preceding unsigned comment added by Thisisvikasv (talk • contribs) 12:39, 13 April 2007 (UTC)

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)

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)

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)

An equilateral triangle is non-embeddable? Cool! Dedhert.Jr (talk) 05:35, 10 August 2026 (UTC)

This was almost certainly known before 1992 (e.g. JSTOR 27964661 from 1985 discusses this, and JSTOR 3615349 from from 1973 has a proof for all regular n-gons with odd n, citing JSTOR 3613163 from 1970 about the equilateral triangle case), but you'd need someone with comprehensive knowledge of old (19th century? 18th century?) math journals to track down the earliest examples. –jacobolus (t) 05:51, 10 August 2026 (UTC)

See also doi:10.1080/00029890.1963.11987605 (1963, "Polygon Imbedded in a Lattice", p. 447) and doi:10.1080/0025570X.2002.11953101 (1992). –jacobolus (t) 06:20, 10 August 2026 (UTC)

One paper cites W. Scherrer, "Die Einlagerung eines Regularen Vielecks in ein Gitter", Elemente der Math., 1 (1946) 97–98. –jacobolus (t) 06:07, 10 August 2026 (UTC)

https://eudml.org/doc/140442 for anyone who reads German (not me). It might be about the related theorem that to have coordinates in any lattice in any dimension a regular polygon can only be an equilateral triangle, square, or regular hexagon. —David Eppstein (talk) 06:27, 10 August 2026 (UTC)

Well, that might be a good exercise for me personally to learn a new language. Dedhert.Jr (talk) 07:34, 10 August 2026 (UTC)

There goes my first guess at your first language. —Antonissimo (talk) 04:53, 11 August 2026 (UTC)

File:Triangle.Equilateral.svg

The area is given as sqrt(3)/4 * 'a'^2. What is 'a'? There is no reference to 'a' prior to this occurrence. TorlachRush (talk) 23:23, 22 October 2024 (UTC)

Sure there is: "An equilateral triangle is a triangle …" Just kidding. a is the side length.

Why link the image as the section title? —Tamfang (talk) 00:18, 24 October 2024 (UTC)

$a$ denotes the magnitude of the side length of an equilateral triangle, which acceptable for counting them according to WP:CALC. Dedhert.Jr (talk) 05:42, 24 October 2024 (UTC)

packing

A packing problem asks the objective of $n$ circles packing into the smallest possible equilateral triangle. The optimal solutions show $n<13$ that can be packed into the equilateral triangle, but the open conjectures expand to $n<28$.

I think this means that proven optima are known for all n≤12 (confirmed here), but surely there are "best known" solutions for some n≥28.

I made many tweaks for flow or for better English. —Tamfang (talk) 07:03, 28 October 2024 (UTC)

Thanks. That summary was taken from the lead of the article Circle packing in an equilateral triangle. Dedhert.Jr (talk) 10:41, 28 October 2024 (UTC)

Nonsense

"The cevians of an equilateral triangle are all equal in length, resulting in the median and angle bisector being equal in length, considering those lines as their altitude depending on the base's choice." A cevian is a line segment from one vertex to the opposite side. In the diagram immediately below (an equilateral triangle of side 2), AM = sqrt(3) and AC = 2 and these are manifestly not equal. Also, the median and angle bisector are equal because they coincide; no waffle about "considering them as an altitude" is needed to prove this. 2A00:23C7:5486:EC01:9B5:E977:9AD8:C38D (talk) 21:18, 29 March 2025 (UTC)

Where does the quoted sentence suggest that the side and altitude are equal? —Tamfang (talk) 05:03, 30 March 2025 (UTC)

The statement "The cevians of an equilateral triangle are all equal in length" is nonsensical because "the cevians of an equilateral triangle" are an infinite family of line segments with lengths ranging between 2 and √3, to take the example of a triangle with side length 2. –jacobolus (t) 05:27, 30 March 2025 (UTC)