MYSTERIES OF THE EQUILATERAL TRIANGLE - HIKARI Ltd

36 Mathematical Properties
Figure 2.12: Fermat Point
connecting each vertex of the original triangle to the new vertex of the opposite
equilateral triangle. These three segments are of equal length and intersect at
the isogonic center where they are inclined at 60 ◦ to one another. At the
isogonic center, each side of the original triangle subtends an angle of 120 ◦ .
Also, the isogonic center lies at the common intersection of the circumcircles of
the three equilateral triangles [322]. The algebraic sum of the distances from
the isogonic center to the vertices of the triangle equals the length of the equal
segments from the latter to the opposite vertices of the equilateral triangles
[189].
Property 13 (Largest Circumscribed Equilateral Triangle). If we connect
the isogonic center of an arbitrary triangle with its vertices and draw lines
through the latter perpendicular to the connectors then these lines intersect to
form the largest equilateral triangle circumscribing the given triangle [189].
This is the antipedal triangle of the isogonic center with respect to the given
triangle.
Property 14 (Napoleon’s Theorem). On each side of an arbitrary triangle,
construct an equilateral triangle pointing outwards. The centers of these three
triangles form an equilateral triangle [178] called the outer Napoleon triangle
(Figure 2.13(a)).