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X(7), X(20), X(21), X(27), X(63), X(84), X(8822) vertices of the 1st Conway triangle, see ETC X(7411) pivot X(8822), see below 

The hexyl triangle is the reflection in O of the excentral triangle IaIbIc. See also TCCT p.172. The locus of P whose cevian triangle is perspective to the hexyl triangle is K344. The locus of the perspector is the Darboux cubic. K344 is the pK with pole X81 and pivot the barycentric product of X86 and X329. This point lies on the lines X(7)X(21), X(19)X(27), X(20)X(64), etc. It is now X(8822) in ETC. K344 is anharmonically equivalent to the Thomson cubic. See Table 21. The isogonal transform of K344 is K750 and its isotomic transform is pK(X321, X322). 
