Parabolic Triangles, Poles and Centroid Relations

We investigate affine properties of centroids formed by three points on a parabola together with the polar triangle formed from the tangents. And we make a wide ranging conjecture about the extension of these results to general conics.

Key words: affinegeometry, parabola, centroids, conics

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Steiner’s Hat: a Constant-Area Deltoid Associated with the Ellipse

The Negative Pedal Curve (NPC) of the Ellipse with respect to a boundary point M is a 3-cusp closed-curve which is the affine image of the Steiner Deltoid. Over all M the family has invariant area and displays an array of interesting properties.

Key words: curve, envelope, ellipse, pedal, evolute, deltoid, Poncelet, osculating, orthologic

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Distance Product Cubics

The locus of points that determine a constant product of their distances to the sides of a triangle is a cubic curve in the projectively closed Euclidean triangle plane. In this paper, algebraic and geometric properties of these distance product cubics shall be studied. These cubics span a pencil of cubics that contains only one rational and non-degenerate cubic curve which is known as the Bataille acnodal cubic determined by the product of the actual trilinear coordinates of the centroid of the base triangle. Each triangle center defines a distance product cubic. It turns out that only a small number of triangle centers share their distance product cubic with other centers. All distance product cubics share the real points of inflection which lie on the line at infinity. The cubics' dual curves, their Hessians, and especially those distance product cubics that are defined by particular triangle centers shall be studied.

Key words: triangle cubic, elliptic cubic, rational cubic, trilinear distance, constant product, Steiner inellipse, triangle centers

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Generalized Helices on a Lightlike Cone in 3-Dimensional Lorentz-Minkowski Space

In this paper we provide characterizations and give some properties of generalized helices in 3-dimensional Lorentz-Minkowski space that lie on a lightlike cone. Furthermore, by analyzing their projections, which turn out to be Euclidean or Lorentzian logarithmic spiral, we present their parametrizations. In particular, we also analyze planar generalized helices, that is planar intersections of a lightlike cone.

Key words: Lorentz-Minkowski space, generalized helix, curve of constant slope, lightlike cone

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The Feuerbach Theorem and Cyclography in Universal Geometry

We have another look at the Feuerbach theorem with a view to extending it in an oriented way to finite fields using the purely algebraic approach of rational trigonometry and universal geometry. Our approach starts with the tangent lines to three rational points on the unit circle, and all subsequent formulas involve the three parameters that define them. Tangency of incircles is treated in the oriented setting via a simplified form of cyclography. Some interesting features of the finite field case are discussed.

Key words: Feuerbach theorem, incircles, universal geometry, cyclography, finite fields

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