engleski

On Brocard Points of Harmonic Quadrangle in I_2(R)

In this talk we present several results concerning the geometry of a harmonic quadrangle in the isotropic plane I_2(R). We consider the standard cyclic quadrangle with the circumscribed circle given by y = x^2 and the vertices chosen to be A = (a, a^2), B = (b, b^2), C = (c, c^2), and D = (d, d^2), with a, b, c, d being mutually different real numbers, a < b < c < d. The harmonic quadrangle in the isotropic plane is a standard cyclic quadrangle with a special property: vertices A, B, C, and D are chosen in a way that tangents A and C at the vertices A and C, respectively, intersect in the point incident with BD, and tangents B and D at the vertices B and D, respectively, are intersected in the point incident with AC. We show that there exist a unique point P_1, so-called the first Brocard point, such that the lines P_1A, P_1B, P_1C, and P_1D form equal angles with the sides AB, BC, CD and DA, respectively. Similarly, the second Brocard point is defined as the point such that the lines P_2A, P_2B, P_2C, and P_2D form equal angles with the sides AD, DC, CB, and BA, respectively. We compare the obtained results with their Euclidean counterparts.

isotropic plane ; harmonic quadrangle ; Brocard points

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