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X(1), X(2), X(7), X(21), X(29), X(77), X(81), X(86) A1B1C1 : cevian triangle of X(86) other points below 

K317 is the locus of point P whose cevian triangle is perspective to CP2, the 2nd circumperp triangle (see TCCT, ยง6.22). The locus of the perspector (which is the cevian quotient of P^2 and K) is K318. See also table 6. K317 has the same asymptotic directions as K455 = pK(X2, X319). It meets the circumcircle at the same points as K320 = pK(X593, X261). K317 is the isogonal transform of K362 = pK(X213, X1) and the isotomic transform of K366 = pK(X321, X75). It is anharmonically equivalent to the Thomson cubic. See Table 21. *** K317 is the third pK of the pencil generated by K002 = pK(X6, X2) and K101 = pK(X1, X1). This pencil also contains a nodal cubic K4 with node X(1) and a decomposed cubic which is the union of the line (L) = X(1)X(2) and the circumconic (C) with perspector X(1). The nine base points of this pencil are A, B, C, X(1) counted twice, X(2) and three other points A', B', C'. The tangents at A', B', C' to K317 concur at X = 5a^2 + 7ab + 7 ac + 8 bc : : . The tangents at A', B', C' to (C) form a triangle whose vertices A", B", C" lie on K002. The inconic (C') with perspector X(86) is also inscribed in A'B'C' and the points of tangency A2, B2, C2 lie on K317. Peter Moses found that A", B", C" lie on the conic (C") which passes through X(i) for i = 5540, 9359, 16554, 24578 and the vertices of the excentral triangle. The perspector of this conic is X(6), and the center is X(36808). Let M be a variable point on (C') with tangent T(M). The pole of T(M) in (C) lies on (C"). All these properties are generalized in Table 72. *** Locus property (Angel Montesdeoca, 20220415) Denote by DEF the cevian triangle of P. IC cuts the perpendicular to IB though D at Ab, and IB cuts the perpendicular to IC though D at Ac. Points Bc, Ba and Ca, Cb are defined cyclically. Let A'B'C' be the triangle formed with lines AbAc, BcBa, CaCb. ABC and A'B'C' are orthologic if and only if P lies on K317. The center of orthology of ABC with respect to A'B'C' lies on K033. 
