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Journal for Geometry and Graphics 21 (2017), No. 2, 201--208
Copyright Heldermann Verlag 2017

Singular Fregier Conics in Non-Euclidean Geometry

Hans-Peter Schroecker
Faculty of Engineering Sciences, University of Innsbruck, Technikerstr. 13, 6020 Innsbruck, Austria

The hypotenuses of all right triangles inscribed into a fixed conic C with fixed right-angle vertex p are incident with the Frégier point f to p and C. As p varies on the conic, the locus of the Frégier point is, in general, a conic as well. We study conics C whose Frégier locus is singular in Euclidean, elliptic and hyperbolic geometry. The richest variety of conics with this property is obtained in hyperbolic plane while in elliptic geometry only three families of conics have a singular Frégier locus.

Keywords: Fregier point, Fregier conic, Thales' theorem, hyperbolic geometry, elliptic geometry, singular conic.

MSC: 51M04; 51M09, 51N35

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