Τετάρτη, 25 Ιανουαρίου 2012

LOCUS


Let ABC be a triangle, A'B'C' the cevian triangle of G, A"B"C" the circumcevian triangle of G with respect the triangle A'B'C' and O1,O2,O3 the circumcenters of GB"C",GC"A",GA"B", resp.


The triangles ABC, O1O2O3 are perspective.

Perspector?

APH, 25 January 2012

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Generalization:

Let ABC be a triangle, A'B'C' the cevian triangle of P, A"B"C" the circumcevian triangle of P with respect the triangle A'B'C' and O1,O2,O3 the circumcenters of PB"C", PC"A", PA"B", resp.
The triangles ABC, O1O2O3 are perspective gives as locus the Yiu quintic and an octic through G and H.

The perspector for H is X381 and that for G is the isotomic conjugate of the point {3 a^4 - 4 a^2 b^2 + b^4 - 4 a^2 c^2 - 6 b^2 c^2 + c^4,
a^4 - 4 a^2 b^2 + 3 b^4 - 6 a^2 c^2 - 4 b^2 c^2 + c^4,
a^4 - 6 a^2 b^2 + b^4 - 4 a^2 c^2 - 4 b^2 c^2 + 3 c^4}, not in ETC.

Francisco Javier García Capitán
26 January 2012

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