The Brocard and Tucker Circles of a Cyclic Quadrilateral

Abstract

1. The application of the geometrical properties of the Brocard and Tucker circles of a triangle to a quadrilateral appears never to have been adequately worked out, as far as the author can discover. Hence, the object of this paper.Some of the problems involved have been published, under the author's name, as independent questions for solution, and where, in the author's opinion, solutions other than his own have seemed more satisfactory for the logical treatment of the subject, these solutions have been employed, with due acknowledgments to their authors. Condition for Brocard poinU.We shall, first establish the condition necessary for the existence of Brocard points within a quadrilateral.Let ABCD (Fig. 1)

On Brocard Points of Harmonic Quadrangle in Isotropic Plane

“…Theorem 9 Let P i 1 and P i 2 be Brocard points of the quadrangle A i B i C i D i , i = 1, 2. Then the second Brocard point P 1 2 of the quadrangle A 1 B 1 C 1 D 1 coincides with P 1 , while the first Brocard point P 2 1 of the quadrangle A 2 B 2 C 2 D 2 coincides with P 2 .…”

“…The Euclidean case of Theorem 2 can be found in [1], and the Euclidean analogue of Theorem 3 is stated in [2] and [4].…”

“…According to [3] (Theorem 4, for h = −2k), the line AP 1 has the equation y = (a + b + 2k)x − a(2k + b). Hence, A 1 has coordinates b + 2k, (b + 2k) 2 . Similarly, we get the other intersections:…”

“…, sides (1) and tangents of its circumscribed circle y = x 2 at its vertices are given by (2). These are the equivalent statements:…”