Editing Equilateral triangle (section)

=== Other mathematical properties ===
[[File:Viviani_theorem_visual_proof.svg|thumb|Visual proof of Viviani's theorem]]
[[Morley's trisector theorem]] states that, in any triangle, the three points of intersection of the adjacent [[angle trisection|angle trisectors]] form an equilateral triangle.

[[Viviani's theorem]] states that, for any interior point <math>P</math> in an equilateral triangle with distances <math>d</math>, <math>e</math>, and <math>f</math> from the sides and altitude <math>h</math>, <math display="block">d+e+f = h,</math> independent of the location of <math>P</math>.{{sfnp|Posamentier|Salkind|1996}}

An equilateral triangle may have [[Integer triangle|integer sides]] with three rational angles as measured in degrees,{{sfnp|Conway|Guy|1996|p=201, 228&ndash;229}} known for the only acute triangle that is similar to its [[orthic triangle]] (with vertices at the feet of the [[altitude (geometry)|altitudes]]),{{sfnp|Bankoff|Garfunkel|1973|p=19}} and the only triangle whose [[Steiner inellipse]] is a circle (specifically, the incircle). The triangle of the largest area of all those inscribed in a given circle is equilateral, and the triangle of the smallest area of all those circumscribed around a given circle is also equilateral.{{sfnp|Dörrie|1965|p=379&ndash;380}} It is the only regular polygon aside from the [[square]] that can be [[inscribed]] inside any other regular polygon.

Given a point <math> P </math> in the interior of an equilateral triangle, the ratio of the sum of its distances from the vertices to the sum of its distances from the sides is greater than or equal to 2, equality holding when <math>P</math> is the centroid. In no other triangle is there a point for which this ratio is as small as 2.{{sfnp|Lee|2001}} This is the [[Erdős–Mordell inequality]]; a stronger variant of it is [[Barrow's inequality]], which replaces the perpendicular distances to the sides with the distances from <math>P</math> to the points where the [[angle bisector]]s of <math>\angle APB</math>, <math>\angle BPC</math>, and <math>\angle CPA</math> cross the sides (<math>A</math>, <math>B</math>, and <math>C</math> being the vertices). There are numerous other [[list of triangle inequalities|triangle inequalities]] that hold equality if and only if the triangle is equilateral.