Parabola-Inscribed Poncelet Derived from the Bicentric Family

We study loci and properties of a Parabola-inscribed family of Poncelet polygons whose caustic is a focus-centered circle. This family is the polar image of a special case of the bicentric family with respect to its circumcircle. We describe closure conditions, curious loci, and new conserved quantities.

Key words: Poncelet, closure, porism, parabola, bicentric, conservation, invariants

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Triads of Conics Associated with a Triangle

We revisit constructions based on triads of conics with foci at pairs of vertices of a reference triangle. We nd that their 6 vertices lie on well-known conics, whose type we analyze. We give conditions for these to be circles and/or degenerate. In the latter case, we study the locus of their center.

Key words: triangle, conic, Carnot, Soddy circles

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Pencil of Frégier Conics

For each point P on a conic c, the involution of right angles at P induces an elliptic involution on c whose center F is called the Frégier point of P. Replacing the right angles at P between assigned pairs of lines with an arbitrary angle f yields a projective mapping of lines in the pencil about P, and thus, on c. The lines joining corresponding points on c do no longer pass through a single point and envelop a conic f which can be seen as the generalization of the Frégier point and shall be called a generalized Frégier conic. By varying the angle, we obtain a pencil of generalized Frégier conics which is a pencil of the third kind. We shall study the thus defined conics and discover, among other objects, general Poncelet triangle families.

Key words: conic, angle, projective mapping, Frégier point, Frégier conic, Poncelet porism, envelope

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On A Triple of Projective Billiards

A projective billiard is a polygon in the real projective plane with a circumconic and an inconic. Similar to the classical billiards in conics, the intersection points between the extended sides of a projective billiard are located on a family of conics which form the associated Poncelet grid. We extend the projective billiard by the inner and outer billiard and disclose various relations between the associated grids and the diagonals, in particular other triples of projective billiards.

Key words: ellipse, billiard, caustic, Poncelet grid, billiard motion

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János Bolyai’s Angle Trisection Revisited

J. Bolyai proposed an elegant recipe for the angle trisection via the intersection of the arcs of the unit circle with that of an equilateral hyperbola c. It seems worthwhile to investigate the geometric background of this recipe and use it as the basic idea for finding the n^th part of a given angle. In this paper, we shall apply this idea for the trivial case n = 4, and for 5. Following Bolyai in the case 5, one has to intersect the unit circle with cubic curve c. There, and in the cases n ≥5, we find only numerical solutions, which shows the limitation of Bolyai's method. Therefore, we propose another construction based on epicycloids inscribed to the unit circle. By this method is even possible to construct the (n/m )^th part of a given angle.

Key words: angle trisection, angle n-section, equilateral hyperbola, cubic, epicycloid

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A complete quadrilateral in rectangular coordinates

A complete quadrilateral in the Euclidean plane is studied. The geometry of such quadrilateral is almost as rich as the geometry of a triangle, so there are lot of associated points, lines and conics. Hereby, the study was performed in the rectangular coordinates, symmetrically on all four sides of the quadrilateral with four parameters a, b, c, d. In this paper we will study the properties of some points, lines and circles associated to the quadrilateral. All these properties are well known, but here they are all proved by the same method. During this process, still some new results have appeared.

Key words: Euclidean plane, complete quadrilateral, parabola

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