A Family of Eight-Point Conics Associated with the Cyclic Quadrilateral
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arXiv
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| Autori principali: | , , , |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866915812783161344 |
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| author | Chomicz, Kazimierz Płatek, Miłosz Smolira, Konstanty Wyrzykowski, Dylan |
| author_facet | Chomicz, Kazimierz Płatek, Miłosz Smolira, Konstanty Wyrzykowski, Dylan |
| contents | We consider the following configuration. Let $ABCD$ be a cyclic quadrilateral with circumcenter $O$, and for each vertex $X$, let $H_X$ be the orthocenter of the triangle formed by the other three. Then $A,\;B,\;C,\;D,\;H_A,\;H_B,\;H_C,\;H_D$ all lie on a single conic. In this paper we study a certain generalization of this fact as follows. For an arbitrary point $P_D$ on the Euler line of $\triangle ABC$, we define corresponding points $P_A, P_B, P_C$ on the respective Euler lines such that the ratio $P_XH_X : P_XO$ is constant for all $X$. We show that the four vertices $A,B,C,D$ and the four isogonal conjugates $Q_A,\;Q_B\;,Q_C\;,Q_D$ of the points $P_X$ all lie on a single conic. This result is given distinct treatments, synthetic, projective, and algebraic. Furthermore, we situate the points $P_X$ within the list of triangle centers. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2511_13298 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A Family of Eight-Point Conics Associated with the Cyclic Quadrilateral Chomicz, Kazimierz Płatek, Miłosz Smolira, Konstanty Wyrzykowski, Dylan Metric Geometry Primary 51M04, 51M05, 51M15 We consider the following configuration. Let $ABCD$ be a cyclic quadrilateral with circumcenter $O$, and for each vertex $X$, let $H_X$ be the orthocenter of the triangle formed by the other three. Then $A,\;B,\;C,\;D,\;H_A,\;H_B,\;H_C,\;H_D$ all lie on a single conic. In this paper we study a certain generalization of this fact as follows. For an arbitrary point $P_D$ on the Euler line of $\triangle ABC$, we define corresponding points $P_A, P_B, P_C$ on the respective Euler lines such that the ratio $P_XH_X : P_XO$ is constant for all $X$. We show that the four vertices $A,B,C,D$ and the four isogonal conjugates $Q_A,\;Q_B\;,Q_C\;,Q_D$ of the points $P_X$ all lie on a single conic. This result is given distinct treatments, synthetic, projective, and algebraic. Furthermore, we situate the points $P_X$ within the list of triangle centers. |
| title | A Family of Eight-Point Conics Associated with the Cyclic Quadrilateral |
| topic | Metric Geometry Primary 51M04, 51M05, 51M15 |
| url | https://arxiv.org/abs/2511.13298 |