A quadrilateral half-turn theorem
Fuente:
arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866917969523638272 |
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| author | Minevich, Igor Morton, Patrick |
| author_facet | Minevich, Igor Morton, Patrick |
| contents | If $ABC$ is a given triangle in the plane, $P$ is any point not on the extended sides of $ABC$ or its anticomplementary triangle, $Q$ is the complement of the isotomic conjugate of $P$ with respect to $ABC$, $DEF$ is the cevian triangle of $P$, and $D_0$ and $A_0$ are the midpoints of segments $BC$ and $EF$, respectively, a synthetic proof is given for the fact that the complete quadrilateral defined by the lines $AP, AQ, D_0Q, D_0A_0$ is perspective by a Euclidean half-turn to the similarly defined complete quadrilateral for the isotomic conjugate $P'$ of $P$ . This fact is used to define and prove the existence of a generalized circumcenter and generalized orthocenter for any such point $P$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_22073 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A quadrilateral half-turn theorem Minevich, Igor Morton, Patrick Algebraic Geometry 51A05 If $ABC$ is a given triangle in the plane, $P$ is any point not on the extended sides of $ABC$ or its anticomplementary triangle, $Q$ is the complement of the isotomic conjugate of $P$ with respect to $ABC$, $DEF$ is the cevian triangle of $P$, and $D_0$ and $A_0$ are the midpoints of segments $BC$ and $EF$, respectively, a synthetic proof is given for the fact that the complete quadrilateral defined by the lines $AP, AQ, D_0Q, D_0A_0$ is perspective by a Euclidean half-turn to the similarly defined complete quadrilateral for the isotomic conjugate $P'$ of $P$ . This fact is used to define and prove the existence of a generalized circumcenter and generalized orthocenter for any such point $P$. |
| title | A quadrilateral half-turn theorem |
| topic | Algebraic Geometry 51A05 |
| url | https://arxiv.org/abs/2503.22073 |