MYSTERIES OF THE EQUILATERAL TRIANGLE - HIKARI Ltd

48 Mathematical Properties
Figure 2.25: Equilic Quadrilateral [181]
Property 33 (Equilic Quadrilaterals). With reference to Figure 2.25, a
quadrilateral ABCD is equilic if AD = BC and ∠A + ∠B = 120 ◦ . Figure
2.26(a): The midpoints P, Q and R of the diagonals and the side CD always
determine an equilateral triangle. Figure 2.26(b): If an equilateral triangle
PCD is drawn outwardly on CD then ∆PAB is also equilateral [181].
(a)
Figure 2.26: (a) Equilic Midpoints. (b) Equilic Triangles. [181]
Property 34 (The Only Rational Triangle). If a triangle has side lengths
which are all rational numbers and angles which are all a rational number of
degrees then the triangle must be equilateral [55]!
Property 35 (Six Triangles). From an arbitrary point in an equilateral
triangle, segments to the vertices and perpendiculars to the sides partition the
triangle into six smaller triangles A, B, C, D, E, F (see Figure 2.27 left).
Claim [183]:
A + C + E = B + D + F.
(b)