KoG, Vol. 29 No. 29, 2025.
Original scientific paper
https://doi.org/10.31896/k.29.7
Projective Parallelians and Related Porisms
Boris Odehnal
orcid.org/0000-0002-7265-5132
; University of Applied Arts Vienna, Vienna, Austria
*
* Corresponding author.
Abstract
We give a projective generalization of the constructionof parallelians and the thus defined conics. To any properly chosen point P and line g in the plane of a triangle Δ = ABC, we construct six points that always lie on a conic P, the parallelian conic P of the pivot P with respect to Δ. Further, we find the parallelian tangent conic T , the parallelian inconic I , and two further conics D and J that are related in a natural way with Δ and P. Any pair out of these conics gives rise to a certain porism and even a chain of porisms by means of polarization. We study the regularity and singularity as well as the relative position of these conics with respect to the line g depending on the choice of P and g. We also give a detailed study of the sets of possible pivot points changing the triangle or hexagon porisms of any pair of conics into such with one-parameter families of quadrangles and pentagons.
Keywords
parallelian; parallelian conic; porism; triangle cubic,; triangle center; algebraic transformation
Hrčak ID:
341894
URI
Publication date:
19.12.2025.
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