On Brocard Points of Harmonic Quadrangle in Isotropic Plane

Abstract

U radu se prikazuju neki novi rezultati o Brocardovim točkama harmoničnog četverokuta u izotropnoj ravnini. Konstruiraju se novi harmonični četverokuti pridruženi danom četverokutu, te se proučavaju njihova svojstva vezana uz Brocardove točke.

Curves of Brocard Points in Triangle Pencils in Isotropic Plane

“…This fact can be proved by putting y = m(x − a+b 2 ) + ab, m ∈ R, into (13), which than becomes an equation in x with a triple root x = a+b 2 . Only in the special case when y = 2bx − b 2 , the equation 13takes the form (a − b)(a + b − 2x) 5 = 0 and x = a+b 2 is its fivefold root. Thus, all tangents to k t1 at C t coincide with t B , Figure 4.…”

“…In [1] the author gave a historical overview and presented many results regarding the Brocard points of polygons in the Euclidean plane. The Brocard points of the triangles in the isotropic plane were introduced and studied in [3], [7] and [8], while such points for harmonic quadrangles were observed in [5] and [6]. In this paper we study the pencil of triangles having the same circumscribed circle and determine the locus of their Brocard points.…”

Bisectors of conics in the isotropic plane

“…According to their position with respect to the absolute figure, conics are classified into ellipses, hyperbolas, special hyperbolas, parabolas and circles, as explained in [2] and [5]. An ellipse is a conic that intersects the absolute line in a pair of complex conjugate points, while a hyperbola intersects it in two different real points.…”