Poncelet triangles: conic loci of the orthocenter and of the isogonal conjugate of a fixed point
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| Format: | Preprint |
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2025
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| _version_ | 1866915443238764544 |
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| author | Garcia, Ronaldo A. Helman, Mark Reznik, Dan |
| author_facet | Garcia, Ronaldo A. Helman, Mark Reznik, Dan |
| contents | We prove that over a Poncelet triangle family interscribed between two nested ellipses $\mathcal{E},\mathcal{E}_c$, (i) the locus of the orthocenter is not only a conic, but it is axis-aligned and homothetic to a $90^o$-rotated copy of $\mathcal{E}$, and (ii) the locus of the isogonal conjugate of a fixed point $P$ is also a conic (the expected degree was four); a parabola (resp. line) if $P$ is on the (degree-four) envelope of the circumcircle (resp. on $\mathcal{E}$). We also show that the envelope of both the circumcircle and radical axis of incircle and circumcircle contain a conic component if and only if $\mathcal{E}_c$ is a circle. The former case is the union of two circles! |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2508_02368 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Poncelet triangles: conic loci of the orthocenter and of the isogonal conjugate of a fixed point Garcia, Ronaldo A. Helman, Mark Reznik, Dan Metric Geometry Graphics 51M04, 51N20, 51N35, 68T20 We prove that over a Poncelet triangle family interscribed between two nested ellipses $\mathcal{E},\mathcal{E}_c$, (i) the locus of the orthocenter is not only a conic, but it is axis-aligned and homothetic to a $90^o$-rotated copy of $\mathcal{E}$, and (ii) the locus of the isogonal conjugate of a fixed point $P$ is also a conic (the expected degree was four); a parabola (resp. line) if $P$ is on the (degree-four) envelope of the circumcircle (resp. on $\mathcal{E}$). We also show that the envelope of both the circumcircle and radical axis of incircle and circumcircle contain a conic component if and only if $\mathcal{E}_c$ is a circle. The former case is the union of two circles! |
| title | Poncelet triangles: conic loci of the orthocenter and of the isogonal conjugate of a fixed point |
| topic | Metric Geometry Graphics 51M04, 51N20, 51N35, 68T20 |
| url | https://arxiv.org/abs/2508.02368 |