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K1343

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X(3), X(4), X(5), X(49), X(54), X(1147), X(3432), X(10274), X(58923), X(58924), X(58925)

imaginary foci of the MacBeath inconic

infinite points of pK(X6,X58922) where X(58922) is the reflection of X(49) in X(5)

points of pK(X6,X49) on (O)

vertices of the Kosnita triangle OaObOc

A1 = BC /\ X(5)Oa, B1 and C1 likewise

Geometric properties :

K1343 is a MacBeath cubic as in Table 80.

The vertices Oa, Ob, Oc of the Kosnita triangle are the circumcenters of the triangles BOC, COA, AOB. See K388, K389, Q064 and Table 31.

The remaining points of K1343 on the perpendicular bisectors of ABC are Qa, Qb, Qc.

Qa = a^4 : (a c + b^2 - c^2) (a c - b^2 + c^2) : (a b + b^2 - c^2) (a b - b^2 + c^2), and Qb, Qc likewise.

This triangle QaQbQc is perspective to :

• the cevian triangles of X(94), X(324) (A'B'C' on the figure), X(14165), X(18883) at X(49), X(10274), X(4), X(5) respectively, and more generally the cevian triangle of every point on a circum-cubic passing through X(2), X(94), X(324), X(14165), X(18883), X(40427), the perspector lying on K1343.

• the anticevian triangles of X(50), X(571), X(16310) at X(49), X(10274), X(5) respectively, and more generally the anticevian triangle of every point on a circum-cubic passing through X(6), X(50), X(571), X(2165), X(14910), X(16310), the perspector again lying on K1343.

• the circum-cevian triangles of X(4), X(94) at X(11464), X(110) respectively, and more generally the circum-cevian triangle of every point on a circum-quartic passing through X(1), X(2), X(4), X(94) and the excenters.

***

Remark : every spK(P, X5) with P on the line (L) through X(5), X(49) passes through X(5). This is the case of K005, K526, K1180 and K1343.

This line (L) contains X(i) for i = 5, 49, 54, 110, 265, 567, 1141, 3615, 4993, 6288, 7604, 8254, 8836, 8838, 8901, 9705, 9706, 10211, 10272, 11597, 11801, 11804, 12022, 12026, 12228, 13434, 14389, 14516, 14643, 14644, 14674, 14769, 15089, 15367, 15425, 15426, 15806, 18350, 18464, 18883, 19176, 19193, 20584, 20585, 23236, 25339, 27196, 27423, 31656, 31675, 32410, 32423, 32638, 34308, 34596, 34597, 34770, 36966, etc.

***

Additional properties of triangle QaQbQc
K1343ped

QaQbQc is homothetic to the pedal triangle A'B'C' of X, on the lines {3,323}, {30,94}, {50,74}, SEARCH = 7.66369983874606.

X = a^2 (2 a^4-4 a^2 b^2+2 b^4-a^2 c^2-b^2 c^2-c^4) (2 a^4-a^2 b^2-b^4-4 a^2 c^2-b^2 c^2+2 c^4) (a^4 b^2-2 a^2 b^4+b^6+a^4 c^2+2 a^2 b^2 c^2-b^4 c^2-2 a^2 c^4-b^2 c^4+c^6) : : , barycentric quotient X(3580)÷X(X44135).

The homothetic center Y lies on the lines {3,323}, {50,194}, SEARCH = 13.8736893051380.

Y = a^4 (2 a^4-4 a^2 b^2+2 b^4-a^2 c^2-b^2 c^2-c^4) (2 a^4-a^2 b^2-b^4-4 a^2 c^2-b^2 c^2+2 c^4) : : , isogonal conjugate of X(44135).

These points X, Y are now X(58940), X(58941) in ETC.
K1343apd

QaQbQc is homothetic to the antipedal triangle A'B'C' of X', on the lines {4,110}, {30,50}, {74,94}, SEARCH = -11.2977518046225.

X' = (a^4+a^2 b^2-2 b^4+a^2 c^2+4 b^2 c^2-2 c^4) (a^6-a^4 b^2-a^2 b^4+b^6-2 a^4 c^2+2 a^2 b^2 c^2-2 b^4 c^2+a^2 c^4+b^2 c^4) (a^6-2 a^4 b^2+a^2 b^4-a^4 c^2+2 a^2 b^2 c^2+b^4 c^2-a^2 c^4-2 b^2 c^4+c^6) : : , barycentric product X(381) x X(2986).

The homothetic center Y' lies on the lines {4,74}, {94,110}, SEARCH = 0.104488634235674.

Y' = -a^12+2 a^10 b^2-2 a^6 b^6+a^4 b^8+2 a^10 c^2-5 a^8 b^2 c^2+3 a^6 b^4 c^2+a^4 b^6 c^2+a^2 b^8 c^2-2 b^10 c^2+3 a^6 b^2 c^4-4 a^4 b^4 c^4-a^2 b^6 c^4+8 b^8 c^4-2 a^6 c^6+a^4 b^2 c^6-a^2 b^4 c^6-12 b^6 c^6+a^4 c^8+a^2 b^2 c^8+8 b^4 c^8-2 b^2 c^10 : : .

Note that X(3), X', Y' are collinear.

These points X', Y' are now X(58942), X(58943) in ETC.