MYSTERIES OF THE EQUILATERAL TRIANGLE - HIKARI Ltd

50 Mathematical Properties
Figure 2.29: Random Point [177]
Property 37 (Random Point). A point P is chosen at random inside an
equilateral triangle. Perpendiculars from P to the sides of the triangle meet
these sides at points X, Y , Z. The probability that a triangle with sides PX,
PY , PZ exists is equal to 1
4 [177].
As shown in Figure 2.29, the segments satisfy the triangle inequality if and
only if the point lies in the shaded region whose area is one fourth that of the
original triangle [122]. Compare this result to Pompeiu’s Theorem!
Property 38 (Gauss Plane). In the Gauss (complex) plane [81], ∆ABC is
equilateral if and only if
(b − a)λ 2 ± = (c − b)λ± = a − c; λ± := (−1 ± ı √ 3)/2.
For λ±, ∆ABC is described counterclockwise/clockwise, respectively.
Property 39 (Gauss’ Theorem on Triangular Numbers). In his diary
of July 10, 1796, Gauss wrote [290]:
“EΥPHKA! num = ∆ + ∆ + ∆.”
I.e., “Eureka! Every positive integer is the sum of at most three triangular
numbers.”
As early as 1638, Fermat conjectured much more in his polygonal number
theorem [319]: “Every positive integer is a sum of at most three triangular
numbers, four square numbers, five pentagonal numbers, and n n-polygonal
numbers.” (Alas, his margin was once again too narrow to hold his proof!)
Jacobi and Lagrange proved the square case in 1772, Gauss the triangular case
in 1796, and Cauchy the general case in 1813 [320].