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X₃₄₁₅–X₃₅₁₄¶

Vertex Conjugates, Saragossa Points & Neuberg Cubic Intersections

U-vertex conjugates of X formed from circumcevian vertex triangles (X₃₄₁₅–X₃₄₅₈), 1st Saragossa points (X₃₄₄₉–X₃₄₅₄), X₅- and X₃₀-Ceva conjugates on the Neuberg cubic (X₃₄₆₀–X₃₄₈₈), and the 1st dilation center and its circumcircle inverse (X₃₅₁₃–X₃₅₁₄).

X₃₄₁₅–X₃₅₁₄ (Vertex Conjugates, Saragossa Points & Neuberg Cubic Intersections) — 16 representative centers (from 100 in collection) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).¶

Group Preamble — Vertex Conjugates

Suppose that U =\(u : v : w\) and X =\(x : y : z\) (trilinears) are points not on a sideline of ABC. Let

\(f(a,b,c) = a/[a^{2}vwyz - ux(bw + cv)(bz + cy)]\).

The U-vertex conjugate of X is the point

\(f(a,b,c) : f(b,c,a) : f(c,a,b)\).

For a geometric interpretation, let T be the vertex triangle of the circumcevian triangles, A_UB_UC_U and A_XB_XC_X, of U and X; viz., the sidelines of T are A_UA_X, B_UB_X, C_UC_X. Then T is perspective to ABC, and the perspector is the U-vertex conjugate of X.

The definition of vertex conjugate allows X = U. To extend the geometric interpretation to the case that X = U, as X approaches U, the vertex triangle approaches a limiting triangle which we call the tangential triangle of U, a triangle perspective to ABC with perspector U-vertex conjugate of U.

The appearance of a row I, J, K in the following tables signifies that the X_i-vertex conjugate of X_j is X(K).

I J K 1 1 56 1 2 3415 1 3 84 1 4 3417 1 6 2163 1 7 3418 1 9 3420 1 19 3422 1 56 1 1 57 3423 1 58 58 2 2 25 2 3 3424 2 4 3425 2 6 1383 2 32 3407 2 251 251 2 523 23 3 3 64 3 4 4 3 6 3426 3 7 3427 3 20 3346 3 40 3345 3 56 945 3 64 3 3 84 1 3 1490 3447 3 1498 3348 3 2131 3183 3 3182 3354 4 4 3 5 5 3432 6 6 6 7 7 3433 8 8 3434 9 9 1436 10 10 3437

Among properties of vertex conjugation are these:

X₃-vertex conjugation maps the Darboux cubic to the Darboux cubic. The appearance of (i,j) in the following list means that X_i is on the Darboux cubic and that X_j = X₃-vertex conjugate of X_i:

(1,84), (3,64), (4,4), (20,3346), (40,3345), (1490,3347), (1498,3348), (2131,3183), (3182,3354)

The fixed point of U-vertex conjugation is the 1st Saragossa point of U. (Saragossa points are defined just before X₁₁₆₆.) The appearance of (i,j) in the following list means that the 1st Saragossa point of X_i is X_j:

(1,58), (2,251), (3,4), (4,54), (5,1166), (6,6), (7,3449), (8,3450), (9,3451), (21,961), (55,57)

If U' and X' are the isogonal conjugates of U and X, resp., then the isogonal conjugate of the U-vertex conjugate of X is the anticomplement of the midpoint of U' and X'. (Randy Hutson, July 11, 2019)

The U-vertex conjugate of U is the isogonal conjugate of the anticomplement of the isogonal conjugate of U. (Randy Hutson, January 17, 2020)