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X₁₇₀₇–X₁₇₈₈¶

Mimosa Transforms

Centers generated by the Mimosa transform M(X) of classical triangle centers.

X₁₇₀₇–X₁₇₈₈ (Mimosa Transforms) — 16 representative centers (from 82 in collection) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).¶

Group Preamble — Mimosa Transforms 1707-1788 (Mimosa transforms and inverse Mimosa tranforms)

The Mimosa transform M(X) of a point X =\(x : y : z\) is defined (9/15/03) by

M(X) = - yz cos A + zx cos B + xy cos C : - zx cos B + xy cos C + yz cos A : - xy cos C + yz cos A + xy cos C

and the inverse Mimosa transform \(M ^{-1}(X)\), by

\(M ^{-1}(X)\) = (cos A)/(y + z) : (cos B)/(z + x) : (cos C)/(x + y).

As with other names in ETC, the name Mimosa is that of a star. M(X) is the X*X₄-Ceva conjugate of X₁, where * denotes trilinear product, and \(M ^{-1}(X)\) is the trilinear quotient X₃/P(X), where P(X) is the crosssum of X₁ and X.

Let g(P,X) denote the P-gimel conjugate of X. The Mimosa transform M(X) arises in connection with the equation \(g(P,X) = X\). Referring to the definition of gimel conjugate in the Glossary, if

P =\(p : q : r\) and X =\(x : y : z\)

are triangle centers, then the equation \(g(P,X) = X is equivalent to the ratio-equation\)

(2absvS - 8yσ²)/(2abswS - 8zσ²) = y/z,

where v = (cos A)/p - (cos B)/q + (cos C)/r and w = (cos A)/p + (cos B)/q - (cos C)/r.

If S = 0, then the ratio-equation holds. As S = x(bq + cr) + y(cr + ap) + z(ap + bq), it follows that if P is given, then \(g(P,X) = X if X is on the line S\)= 0 (regarding \(x : y : z\) as variable); and that if X is given, then \(g(P,X) = P if P is on the line S\)= 0 (regarding \(p : q : r\) as variable). There are too many such cases of gimel conjugates for all to be itemized in ETC. For example, if X = X₁, then \(g(P,X) = X\) for every P on the line at infinity; if X = X₅₁₃, then \(g(P,X) = X\) for every P on the line X₁ X₂; and if X = X₆₅₆, then \(g(P,X) = X\) for every P on the Euler line, X₂ X₃.

If S ≠ 0, the ratio-equation lends itself to easy simplifications and two Tables conclusions: (1) if P is given then X = M(P) is a solution of \(g(P,X) = X\), and (2) if X is given then P = M*(X) is a solution of \(g(P,X) = X\).

Here is a list of pairs (i,j) for which X_j = M(X_i):

1,46 2,19 3,1 4,920 8,1158 21,4 48,43 54,47 59,109 60,580 63,9 69,63 71,846 72,191 73,1046 77,57 78,40 81,579 90,90 95,92 96,91 97,48 99,1577 110,656 219,165 228,1045 248,1580 249,163 250,162 252,564 254,921 271,84 283,3 348,169 394,610 603,978 648,822 651,652 662,1021 895,896 1105,158 1176,31 1259,1490 1297,240 1332,649 1444,2 1459,1054

Of course, reversing the pairs gives a list of (J,I) for which X_i = M*(X_j).