Editing Reuleaux triangle (section)

== Construction ==
[[File:Construction triangle Reuleaux.svg|thumb|left|upright|To construct a Reuleaux triangle]]
The Reuleaux triangle may be constructed either directly from three [[circles]], or by rounding the sides of an [[equilateral triangle]].<ref name="hann">{{citation|title=Structure and Form in Design: Critical Ideas for Creative Practice|first=Michael|last=Hann|publisher=A&C Black|year=2014|isbn=978-1-4725-8431-1|page=34|url=https://books.google.com/books?id=-CX-AgAAQBAJ&pg=PA34}}.</ref>

The three-circle construction may be performed with a [[Compass (drafting)|compass]] alone, not even needing a straightedge. By the [[Mohr–Mascheroni theorem]]
the same is true more generally of any [[compass-and-straightedge construction]],<ref>{{citation
 | last = Hungerbühler | first = Norbert
 | doi = 10.2307/2974536
 | issue = 8
 | journal = [[American Mathematical Monthly]]
 | mr = 1299166
 | pages = 784–787
 | title = A short elementary proof of the Mohr-Mascheroni theorem
 | volume = 101
 | year = 1994| jstor = 2974536
 | citeseerx = 10.1.1.45.9902
 }}.</ref> but the construction for the Reuleaux triangle is particularly simple.
The first step is to mark two arbitrary points of the plane (which will eventually become vertices of the triangle), and use the compass to draw a circle centered at one of the marked points, through the other marked point. Next, one draws a second circle, of the same radius, centered at the other marked point and passing through the first marked point.
Finally, one draws a third circle, again of the same radius, with its center at one of the two crossing points of the two previous circles, passing through both marked points.<ref>This construction is briefly described by {{harvtxt|Maor|Jost|2014}} and may be seen, for instance, in the video [https://www.youtube.com/watch?v=OdY9Y-6DsgU Fun with Reuleaux triangles] by Alex Franke, August 21, 2011.</ref> The central region in the resulting arrangement of three circles will be a Reuleaux triangle.<ref name="hann" />

Alternatively, a Reuleaux triangle may be constructed from an equilateral triangle ''T'' by drawing three arcs of circles, each centered at one vertex of ''T'' and connecting the other two vertices.<ref name="gardner" />
Or, equivalently, it may be constructed as the intersection of three disks centered at the vertices of ''T'', with radius equal to the side length of ''T''.<ref name="onup" />