Some Remarks on Equilateral Triangles and Squares

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Let a₁, a₂, a₃ denote the sides α₁, α2, α₃ the angles and A₁, A₂, A₃ the vertices of a triangle in the Euclidean plane. A point P, whose distances from the sides α₁, α₂, α₃ are in the ratio p₁: p₂: p₃ will be denoted by P[p₁]. p₁, p₂, p₃ are called the normal coordinates of P. Thus the unit point E[l] is the incentro of the triangle, S[cosec α₁] is the centroid, M[cos α₁] is the circumcentre, and H[sec α₁] is the orthocentre. Similarly, using normal line coordinates, we have l₀ [l] is the unit line, l_∞[sin α₁] is the line at infinity and l_H[cos α₁] is the axis of the altitudes.