MYSTERIES OF THE EQUILATERAL TRIANGLE - HIKARI Ltd

52 Mathematical Properties
Figure 2.32: Largest Inscribed Triangle and Least-Diameter Decomposition of
the Open Disk [4]
Property 42 (Largest Inscribed Triangle). The triangle of largest area
that is inscribed in a given circle is the equilateral triangle (Figure 2.32) [249].
Property 43 (Planar Soap Bubble Clusters). An inscribed equilateral
triangle (Figure 2.32) provides a least-diameter smooth decomposition of the
open unit disk into relatively closed sets that meet at most two at a point [4].
Property 44 (Jung’s Theorem). Let d be the (finite) diameter of a planar
set and let r be the radius of its smallest enclosing circle. Then, [249]
r ≤ d √ 3 .
Since the circumcircle is the smallest enclosing circle for an equilateral triangle
(Figure 2.32), this bound cannot be diminished.
Property 45 (Isoperimetric Theorem for Triangles). Among triangles
of a given perimeter, the equilateral triangle has the largest area [191]. Equivalently,
among all triangles of a given area, the equilateral triangle has the
shortest perimeter [228].
Property 46 (A Triangle Inequality). If A is the area and L the perimeter
of a triangle then
A ≤ √ 3L 2 /36,
with equality if and only if the triangle is equilateral [228].
Property 47 (Euler’s Inequality). If r and R are the radii of the inscribed
and circumscribed circles of a triangle then
R ≥ 2r,
with equality if and only if the triangle is equilateral [191, 228].