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X(1), X(57), X(77), X(81), X(270), X(757), X(2185)

A1B1C1 : cevian triangle of X(757)

A2B2C2 : cevian triangle of X(2185)

other points below

Q173 is analogous to Q172, where general properties are given.

Let PaPbPc be the circumcevian triangle of a variable point P. With I = X(1), let X = BC ∩ IPa, Y = CA ∩ IPb, Z = AB ∩ IPc. ABC and XYZ are perspective (at Q) if and only if P lies on the cubic K318 = pK(X1333, X21), a circumcevian cubic of CL072. The locus of the perspector Q, as P moves on K318, is the sextic Q173.

Q173 is invariant under the isoconjugation with pole X(593), the barycentric square of X(81).

X(81) is a quadruple point on Q173 and A, B, C are double points with tangents passing through X(1) and X(57). The tangents at X(1), A1, B1, C1 pass through X(81).

Points on Q173 :

Q1 = a (a+b-c) (a-b+c) (a^2-b^2-c^2) (a^3-a^2 b-a b^2+b^3+a^2 c+2 a b c+b^2 c-a c^2-b c^2-c^3) (a^3+a^2 b-a b^2-b^3-a^2 c+2 a b c-b^2 c-a c^2+b c^2+c^3) : : , SEACH = 0.950621300618149, on the lines {X7,X84}, {X57,X189}, {X63,X268}, {X69,X271}, {X77,X1433}, {X81,X1422}.

Q2 = a (a+b-c) (a-b+c) (a^2-b^2-c^2) (a^3-a^2 b-a b^2+b^3+a^2 c+2 a b c+b^2 c-a c^2-b c^2-c^3) (a^3+a^2 b-a b^2-b^3-a^2 c+2 a b c-b^2 c-a c^2+b c^2+c^3) (3 a^4-2 a^2 b^2-b^4-2 a^2 c^2+2 b^2 c^2-c^4) : : , SEACH = -0.974562598105507, on the line {X81,X1422}.

Q3 = a (a+b)^2 (a-b-c) (a+c)^2 (a^2+b^2-c^2) (a^2-b^2+c^2) (a^3+a^2 b-a b^2-b^3+a^2 c-2 a b c+b^2 c-a c^2+b c^2-c^3) : : , SEACH = 1.70433925831972, on the lines {X3,X162}, {X28,X60}, {X29,X284}.

Q4 = a (a+b)^2 (a-b-c) (a+c)^2 (a^2+b^2-c^2) (a^2-b^2+c^2) (a^3+a^2 b-a b^2-b^3+a^2 c-2 a b c+b^2 c-a c^2+b c^2-c^3) (a^4-2 a^2 b^2+b^4+2 a^2 c^2+2 b^2 c^2-3 c^4) (a^4+2 a^2 b^2-3 b^4-2 a^2 c^2+2 b^2 c^2+c^4) : : , SEACH = 5.60144226422627, on the line {X81,X2326}.

Q5 = a (a+b)^2 (a+c)^2 (a^2+b^2-c^2) (a^2-b^2+c^2) (a^3+a^2 b-a b^2-b^3+a^2 c-2 a b c+b^2 c-a c^2+b c^2-c^3) (a^4-2 a^2 b^2+b^4+2 a^2 c^2+2 b^2 c^2-3 c^4) (a^4+2 a^2 b^2-3 b^4-2 a^2 c^2+2 b^2 c^2+c^4) (a^6-2 a^5 b-a^4 b^2+4 a^3 b^3-a^2 b^4-2 a b^5+b^6-2 a^5 c-2 a^4 b c+2 a b^4 c+2 b^5 c-a^4 c^2+2 a^2 b^2 c^2-b^4 c^2+4 a^3 c^3-4 b^3 c^3-a^2 c^4+2 a b c^4-b^2 c^4-2 a c^5+2 b c^5+c^6): : , SEACH = 0.678749742514330, on the line {X81,X2326}.

Q6 = a (a^3-a^2 b-a b^2+b^3+a^2 c+2 a b c+b^2 c-a c^2-b c^2-c^3) (a^3+a^2 b-a b^2-b^3-a^2 c+2 a b c-b^2 c-a c^2+b c^2+c^3) (3 a^4-2 a^2 b^2-b^4-2 a^2 c^2+2 b^2 c^2-c^4) (a^6+2 a^5 b-a^4 b^2-4 a^3 b^3-a^2 b^4+2 a b^5+b^6-2 a^5 c+2 a^4 b c+2 a b^4 c-2 b^5 c-a^4 c^2+2 a^2 b^2 c^2-b^4 c^2+4 a^3 c^3+4 b^3 c^3-a^2 c^4-2 a b c^4-b^2 c^4-2 a c^5-2 b c^5+c^6) (a^6-2 a^5 b-a^4 b^2+4 a^3 b^3-a^2 b^4-2 a b^5+b^6+2 a^5 c+2 a^4 b c-2 a b^4 c-2 b^5 c-a^4 c^2+2 a^2 b^2 c^2-b^4 c^2-4 a^3 c^3+4 b^3 c^3-a^2 c^4+2 a b c^4-b^2 c^4+2 a c^5-2 b c^5+c^6) (a^8-4 a^6 b^2+6 a^4 b^4-4 a^2 b^6+b^8-4 a^6 c^2-4 a^4 b^2 c^2+4 a^2 b^4 c^2+4 b^6 c^2+6 a^4 c^4+4 a^2 b^2 c^4-10 b^4 c^4-4 a^2 c^6+4 b^2 c^6+c^8): : , SEACH = 1.26999512613060.

The following table gives the correspondence between P = X(i) on K318 and Q = X(j) on Q173.

P4 = a^2 (a+b) (a-b-c) (a+c) (a^4-2 a^2 b^2+b^4+2 a^2 c^2+2 b^2 c^2-3 c^4) (a^4+2 a^2 b^2-3 b^4-2 a^2 c^2+2 b^2 c^2+c^4) : : , SEARCH = 4.60315330152353, on the lines {X1,X204}, {X3,X64}, {X21,X77}, {X58,X1433}, {X102,X1301}.

P4 is the barycentric quotient X1333 ÷ X1394.