Olga Radko and Emmanuel Tsukerman,
The perpendicular bisector construction, isoptic point and Simson line,
Forum Geometricorum, 12 (2012) 161--189.
Abstract.
Given a noncyclic quadrilateral, we consider an iterative procedure
producing a new quadrilateral at each step. At each iteration, the
vertices of the new quadrilateral are the circumcenters of the triad
circles of the previous generation quadrilateral. The main goal of the paper is to prove a number
of interesting properties of the limit point of this iterative
process. We show that the limit point is the common center of spiral
similarities taking any of the triad circles into another triad
circle. As a consequence, the point has the isoptic property
i.e., all triad circles are visible from the limit point at
the same angle. Furthermore, the limit point can be viewed as a
generalization of a circumcenter. It also has properties similar to
those of the isodynamic point of a triangle. We also characterize
the limit point as the unique point for which the pedal
quadrilateral is a parallelogram. Continuing to study the pedal
properties with respect to a quadrilateral, we show that for every
quadrilateral there is a unique point (which we call the Simson
point) such that its pedal consists of four points on a line, which we
call the Simson line, in analogy to the case of a triangle.
Finally, we define a version of isogonal conjugation for a
quadrilateral and prove that the isogonal conjugate of the limit
point is a parallelogram, while that of the Simson point is a
degenerate quadrilateral whose vertices coincide at infinity.
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