## 21975Re: loci related to Taylor circle

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• Apr 17, 2013
Paul,

This is interesting: the perspector you mention with ETC search value 5.10435062529 matches the isogonal conjugate of the polar conjugate of X(1073), and as such would have trilinears (cos^2 A)/(cos A - cos B cos C) : :.

What do you mean by the notations S_{AB}, etc.? I assume it has to do with Conway notation. Also, are your coordinates trilinears or barycentrics?

Randy

--- In Hyacinthos@yahoogroups.com, "yiuatfauedu" <yiu@...> wrote:
>
> Dear Randy and Bernard,
>
> [RH] Let ABC be a triangle, and P a point.
> Let A'B'C' be the pedal triangle of P.
> Let Ba, Ca be the orthogonal projections of A' onto lines CA, AB, resp.
> Define Cb, Ab, Ac, Bc cyclically.
> What is the locus of P such that Ba, Ca, Cb, Ab, Ac, Bc lie on a common conic? The locus would include H, for which the conic is the Taylor circle.
>
> [BG]: a quintic with many simple points but only two (I think) ETC centers : X4, X1498.
>
> *** X(1498) is the Nagel point of the tangential triangle.
> Ab = (0 : S_{AB}+S_{AC}-S_{BC} : S_{AB}+S_{AC}+S_{BC}) and
> Ac = (0 : S_{AB}+S_{AC}+S_{BC} : S_{AB}+S_{AC}+S_{BC})
> are isotomic points on BC, so are Bc, Ba, and Ca, Cb.
> X(1498) is the unique point with this property. The conic is
>
> (4/S^2)cyclic sum ((a^4S_{AA})/(S_{AB}+S_AC}-S_{BC}))yz - (x+y+z)^2 = 0,
> concentric (and homothetic) with the circumconic with perspector
> ((a^4S_{AA}/(S_{AB}+S_{AC}-S_{BC}):...:...)
> [with (6-9-13)-search number 5.10435062529...]
> and has center
> (a^4(S_{AAAB}+S_{AAAC}+S_{AABB}-S_{AABC}+S_{AACC}-S_{BBCC}/
> (S_{AB}+S_{AC}-S_{BC}) :...:...)
> with (6-9-13) search number 1.09478783248....
>
> Best regards
> Sincerely
> Paul
>
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