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X(3), X(4), X(76), X(7607), X(9716), X(13330), X(14357), X(34507), X(52300), X(67624), X(72811), X(72812), X(72813), X(72814)

imaginary foci of the MacBeath inconic

infinite points of pK(X6, X576)

points of pK(X6, X34507) on (O)

other details and points below

Geometric properties :

K1447 is mentioned by Ivan Pavlov in Euclid #9913. See also Table 80.

K1447 meets the line at infinity at the same points as pK(X6, X576). These two cubics meet again at six points on the circum-conic (C) passing through X(54), X(111), X(7607).

K1447 meets the circumcircle at the same points Q1, Q2, Q3 as pK(X6, X34507). These two cubics meet again at three points on the line (L) passing through X(5), X(524), X(576), X34507).

(C) and (L) are isogonal transform of one another. They meet at M1, M2 which lie on the three cubics above and also on K005.

Recall that the imaginary foci of the MacBeath inconic, the inconic with center X(5) and perspector X(264), lie on the perpendicular bisector of OH and on the cicum-conic passing through X(54), X(110) which is its isogonal transform. The third point Q5 of K1447 on this line also lies on {X(7607), X(52300)} and on {X(34507), X(67624)}.

The parallel at X(34507) to the line through A and X(576) meet (BC) at A' and the line through A and X(7607) at A". These two points A', A" and those obtained cyclically B', B" and C', C" all lie on K1447.

Remarks

X(7607) lies on the Kiepert hyperbola

X(9716) lies on the Thomson-Jerabek hyperbola (see K002) and on the cubic K759

X(13330) lies on the Brocard axis

X(14357) lies on the line {3,67} and on the cubics K009, K072, K284, K1065, K1068, K1369

X(34507) lies on the lines {3,67}, {4,69} and on the cubic K481

X(52300) lies on the Euler line

X(67624) lies on the Jerabek hyperbola

Other centers on K1447

Q1 = {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 - a^2 b^2 c^2 - b^4 c^2 - 2 a^2 c^4 - b^2 c^4 + c^6) : : , SEARCH = -0.853664433606890, on the line {3, 9716}

Q2 = b^2 c^2 (a^4 + b^4 - a^2 c^2 - b^2 c^2) (-a^4 + a^2 b^2 + b^2 c^2 - c^4) (-a^6 + 4 a^4 b^2 - 3 a^2 b^4 + b^6 + 4 a^4 c^2 - a^2 b^2 c^2 - b^4 c^2 - 3 a^2 c^4 - b^2 c^4 + c^6) : : , SEARCH = 3.00756218776948, on the line {3, 76}

Q3 = 4 a^8-16 a^6 b^2+21 a^4 b^4-11 a^2 b^6+2 b^8-16 a^6 c^2+25 a^4 b^2 c^2-4 a^2 b^4 c^2+21 a^4 c^4-4 a^2 b^2 c^4-4 b^4 c^4-11 a^2 c^6+2 c^8 : : , SEARCH = 9.00441774589406, on the lines {3, 67624}, {76,7607}

Q4 = b^2 c^2 (2 a^4-3 a^2 b^2+2 b^4-2 a^2 c^2-2 b^2 c^2) (-2 a^4+2 a^2 b^2+3 a^2 c^2+2 b^2 c^2-2 c^4) (-a^4-a^2 b^2+b^4-a^2 c^2-2 b^2 c^2+c^4) : : , SEARCH = 8.16697215899512, on the line {76,9716}

Q5 = b^2 c^2 (a^2-a b+b^2-c^2) (a^2+a b+b^2-c^2) (-a^2+b^2-a c-c^2) (-a^2+b^2+a c-c^2) (-4 a^6+7 a^4 b^2-5 a^2 b^4+2 b^6+7 a^4 c^2-2 b^4 c^2-5 a^2 c^4-2 b^2 c^4+2 c^6) : : , SEARCH = -0.0412215509250671, on the lines {5,523}, {7607,52300}, {34507,67624}, {3,Q4), {76,Q1}

Q6 = (a^2+b^2-2 c^2) (a^2-2 b^2+c^2) (4 a^10-6 a^8 b^2+2 a^6 b^4+4 a^4 b^6-6 a^2 b^8+2 b^10-6 a^8 c^2+5 a^6 b^2 c^2-12 a^4 b^4 c^2+15 a^2 b^6 c^2-7 b^8 c^2+2 a^6 c^4-12 a^4 b^2 c^4-3 a^2 b^4 c^4+5 b^6 c^4+4 a^4 c^6+15 a^2 b^2 c^6+5 b^4 c^6-6 a^2 c^8-7 b^2 c^8+2 c^10) : : , SEARCH = -1.43073566612551, on the lines {3,7607},{76,67624}

Q7 = a^2 (a^4-b^4+b^2 c^2-c^4) (2 a^4-2 a^2 b^2+5 b^4-5 a^2 c^2-2 b^2 c^2+2 c^4) (2 a^4-5 a^2 b^2+2 b^4-2 a^2 c^2-2 b^2 c^2+5 c^4) : : , SEARCH = 9.112720347672909, on the line {9716,13330}