MYSTERIES OF THE EQUILATERAL TRIANGLE - HIKARI Ltd

38 Mathematical Properties
2.14(b) left/right, construct equilateral triangles pointing outwards/inwards on
the sides of an oriented parallelogram ABCD giving parallelogram XY ZW.
Then, if inward/outward pointing equilateral triangles are drawn on the sides
of oriented parallelogram XY ZW, the resulting parallelogram is just ABCD
again [182].
(a)
Figure 2.15: (a) Isodynamic (Apollonius) Points. (b) Pedal Triangle of First
Isodynamic Point
Property 16 (Pedal Triangles of Isodynamic Points). With reference
to ∆ABC of Figure 2.15(a), let U and V be the points on BC met by the
interior and exterior bisectors of ∠A. The circle having diameter UV is called
the A-Apollonian circle [6]. The B- and C-Apollonian circles are likewise
defined. These three Apollonian circles intersect at the first (J) and second
(J ′ ) isodynamic (Apollonius) points [116]. With reference to Figure 2.15(b),
connecting the feet of the perpendiculars from the first isodynamic point, I ′ ,
to the sides of ∆ABC produces its pedal triangle which is always equilateral
[189]. The same is true for the pedal triangle of the second isodynamic point.
This theorem generalizes as follows: The pedal triangle of any of the four
points A, B, C, I ′ with respect to the triangle formed by the remaining points is
equilateral [46, p. 303].
Property 17 (The Machine for Questions and Answers). In 2006, D.
Dekov created a computer program, The Machine for Questions and Answers,
and used it to produce The Computer-Generated Encyclopedia of Euclidean
Geometry which contains the following results pertinent to equilateral triangles
[73]. (Note: Connecting a point to the three vertices of a given triangle creates
three new triangulation triangles associated with this point. Also, deleting the
cevian triangle [6, p. 160] of a point with respect to a given triangle leaves
three new corner triangles associated with this point.)
(b)