## 19705Re: [EMHL] A point related to a quadrilateral

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• Jan 9, 2011
> Dear Jean-Pierre,
>
> I am quite thrilled.
> As always when I find something beautiful.
> You described a point M depending on the 4 points A1, A2,
> A3, A4 of a quadrilateral. Here is the structure behind it.
>
> M = the inverse in the circumcircle of AiAjAk of the
> isogonal conjugate of Al wrt AiAjAk,
> where (i,j,k,l) is any permutation of (1,2,3,4).
>
> Strangely enough this is true for each permutation.
> There should be more properties/references relating to this point.
> Best regards,
>
> Chris van Tienhoven

I did this using complex numbers as coordinates, where the formula for M should be symmetric in the 4 complex variables (A1,A2,A3,A4), and have some invariance properties. The answer can be expressed as follows:

M is a quotient of two determinants. Each determinant is 4-by-4, and its rows correspond to the variables. In the numerator, the row for x is [1, x, x^2, x x#], where x# means the complex conjugate of x. In the denominator, the row for x is [1, x, x^2, x#].

Can other ETC points be expressed in this way?
--
Barry Wolk
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