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X₁₂₈₆–X₁₃₁₁¶

Collings Transforms

Concurrence points on the circumcircle of the circumcircles of AB'C', A'BC', and A'B'C, where A'B'C' is the reflection of ABC in P.

X₁₂₈₆–X₁₃₁₁ (Collings Transforms) — 14 representative centers (from 26 in collection) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).¶

Group Preamble — Collings Transforms: 1286-1311

If ABC is a triangle, P is a point, and A', B', C' are the reflections of A, B, C in P, then the circumcircles of triangles AB'C', A'BC' and A'B'C concur at a point Q on the circumcircle of triangle ABC. The transformation T given by Q = T(P) was described by

S. N. Collings, “Reflections on reflections 2”, Mathematical Gazette 1974, page 264.

It was further discussed by Floor van Lamoen and Darij Grinberg, and coordinates were found by Barry Wolk; see Hyacinthos #4547, #4548, #6469, #6538, #6546, #6560. Paul Yiu noted that T(P) is the point, other than A, B, C, in which the circumconic centered at P meets the circumcircle of triangle ABC (#4548).

If P =\(x : y : z\) (trilinears), then Q =\(f(a,b,c) : f(b,c,a) : f(c,a,b)\), where

\(f(a,b,c) = 1/[bz(ax + by - cz) - cy(ax + cz - by)]\).

For given Q, the set of points P satisfying T(P) = Q is a conic. Examples follow:

if Q = X74, the conic passes through Xi for I = 125;
if Q = X98, the conic passes through Xi for I = 115, 868;
if Q = X99, the conic passes through Xi for I = 2, 39, 114, 618, 619, 629, 630, 641, 642, 1125;
if Q = X100, the conic passes through Xi for I = 1, 9, 10, 119, 142, 214, 442, 1145;
if Q = X107, the conic passes through Xi for I = 4, 133, 800, 1249;
if Q = X110, the conic passes through Xi for I = 5, 6, 113, 141, 206, 942, 960, 1147, 1209;
if Q = X476, the conic passes through Xi for I = 30.

Bernard Gibert (4/02/03) identified T(P) as the trilinear pole of the line of X₆ and the X₂-Ceva conjugate of P. He identified the locus of P as the rectangular hyperbola that circumscribes the medial triangle and has center W given by the vector equation 4X₂W = X₂Q. The anticomplement of this hyperbola is the rectangular ABC-circumhyperbola whose center is the complement of Q. Thus, referring to examples given above:

if Q = X99, the conic is the Kiepert hyperbola of the medial triangle;
if Q = X100, the conic is the Feuerbach hyperbola of the medial triangle;
if Q = X110, the conic is the Jerabek hyperbola of the medial triangle.