MYSTERIES OF THE EQUILATERAL TRIANGLE - HIKARI Ltd

46 Mathematical Properties
Figure 2.22: Euler Line Cuts Off Equilateral Triangle
Property 28 (Euler Line Cuts Off Equilateral Triangle). If a triangle
is not equilateral then the orthocenter (intersection of altitudes), the centroid
(intersection of medians) and the circumcenter (intersection of perpendicular
bisectors) are collinear [58]. (For an equilateral triangle, these three points
coincide.) Many other important points associated with a triangle, such as the
nine-point center, also lie on this so-called Euler line [6]. In a triangle with a
60 ◦ angle, the Euler line cuts off an equilateral triangle [28] (see Figure 2.22).
Property 29 (Incircle-Triangle Iteration). Let ∆A0B0C0 be arbitrary. Let
the points of contact with its incircle be A1, B1, C1. Let the points of contact of
∆A1B1C1 with its incircle be A2, B2, C2, and so on. This sequence of triangles
shrinks by a factor of 1/2 at each iteration and approaches equiangularity in
the limit [47].
Property 30 (Excentral Triangle Iteration). The bisector of any interior
angle of a triangle and those of the exterior angles at the other two vertices
are concurrent at a point outside the triangle. These three points are called
excenters and they are the vertices of the excentral triangle. Commencing
with an arbitrary triangle, construct its excentral triangle, then construct the
excentral triangle of this excentral triangle, and so on. These excentral triangles
approach an equilateral triangle [189].