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X(2), X(4), X(6), X(8), X(386), X(573), X(941), X(959), X(2345), X(5739), X(11337)

X(34259), X(34258) : isogonal conjugates of X(4185), X(5019)

infinite points of pK(X6, X940)

points of pK(X6, X5739) on the circumcircle

foci of the inconic with center X(5743), the complement of X(940)

other points below

Geometric properties :

K1135 is the isogonal transform of K321 = pK(X5019, X2) hence K1135 = pK(X941, gX5019). Note that the isopivot is X(6) hence the tangents at A, B, C are the symmedians.

K1135 meets the line at infinity at the same points as pK(X6, X940) and the six other common points lie on the circum-hyperbola passing through X(2), X(6), X(941).

K1135 also meets the line at infinity at the same points as pK(X2, Q1) where Q1 = b c (-2 a^3+2 a b^2+a b c+b^2 c+2 a c^2+b c^2) : : , SEARCH = 5.20186120020884, on lines {X4,X69}, {X75,X519}, {X86,X386}, etc. The six other common points lie on the Kiepert hyperbola namely A, B, C, X(2), X(4) and gX(5019).

K1135 meets the Steiner ellipse at the same points as pK(X2, Q2) where Q2 = b c (2 a^3+a b c+b^2 c+b c^2) : : , SEARCH = 3.30056490937275, on lines {X6,X76}, {X9,X1909}, {X75,X527}, etc. The three other common (collinear) points are X(2), X(6), X(5739).

The isotomic transform of K1135 is pK(Q3, X76) where Q3 = b c (a^2+a b+a c+2 b c) : : , SEARCH = 3.55556701126774, on lines {X2,X39}, {X7,X8}, {X21,X1975}, {X86,X964}, etc.

Q1, Q2, Q3 are X(34282), X(34283), X(34284) in ETC.

***

Other centers on K1135 (the isogonal conjugates of these points obviously lie on K321) :

P1 = (a^2 b-b^3+a^2 c-a b c+a c^2+b c^2) (a^3+2 a^2 b+a b^2+2 a^2 c+a b c+b^2 c+a c^2+b c^2) (a^2 b+a b^2+a^2 c-a b c+b^2 c-c^3) : : , SEARCH = -1.37918281838819.

P2 = a (a+b-c) (a-b+c) (a b+b^2+2 a c+b c) (2 a b+a c+b c+c^2) (a^4-b^4+2 a^2 b c-2 a b^2 c-2 a b c^2+2 b^2 c^2-c^4) : : , SEARCH = -0.870917940341744.

P3 = a (a b+b^2+2 a c+b c) (2 a b+a c+b c+c^2) (a^3-a b^2+a b c-b^2 c-a c^2-b c^2) (a^2 b+2 a b^2+b^3+a^2 c+a b c+2 b^2 c+a c^2+b c^2) (a^2 b+a b^2+a^2 c+a b c+b^2 c+2 a c^2+2 b c^2+c^3) : : , SEARCH = 5.14525996964708.

P4 = b c (a^2+a b+b^2+a c+b c) (a b+b^2+2 a c+b c) (a^2+a b+a c+b c+c^2) (2 a b+a c+b c+c^2) : : , SEARCH = 3.49355195567827.

P5 = (a^2+b^2-c^2) (a^2-b^2+c^2) (a^7+a^6 b+a^5 b^2+a^4 b^3-a^3 b^4-a^2 b^5-a b^6-b^7+a^6 c+6 a^5 b c+a^4 b^2 c-4 a^3 b^3 c-a^2 b^4 c-2 a b^5 c-b^6 c+a^5 c^2+a^4 b c^2-2 a^3 b^2 c^2-2 a^2 b^3 c^2+a b^4 c^2+b^5 c^2+a^4 c^3-4 a^3 b c^3-2 a^2 b^2 c^3+4 a b^3 c^3+b^4 c^3-a^3 c^4-a^2 b c^4+a b^2 c^4+b^3 c^4-a^2 c^5-2 a b c^5+b^2 c^5-a c^6-b c^6-c^7) : : , SEARCH = 0.794996948095096.

P6 = (a^2 b-b^3+a^2 c-a b c+a c^2+b c^2) (a^2 b+a b^2+a^2 c-a b c+b^2 c-c^3) (a^6+a^5 b-a^2 b^4-a b^5+a^5 c+3 a^4 b c-2 a^3 b^2 c-2 a^2 b^3 c+a b^4 c-b^5 c-2 a^3 b c^2+2 a^2 b^2 c^2-2 a^2 b c^3+2 b^3 c^3-a^2 c^4+a b c^4-a c^5-b c^5) : : , SEARCH = 0.536719323992364.

P7 = b c (a b+b^2+2 a c+b c) (2 a b+a c+b c+c^2) (a^3+b^3+a b c-a c^2-b c^2) (-a^3+a b^2-a b c+b^2 c-c^3) (-a^6-2 a^5 b+a^4 b^2+2 a^3 b^3-a^2 b^4+b^6-2 a^5 c+2 a b^4 c+a^4 c^2+2 a^2 b^2 c^2-2 a b^3 c^2-b^4 c^2+2 a^3 c^3-2 a b^2 c^3-a^2 c^4+2 a b c^4-b^2 c^4+c^6) : : , SEARCH = 1.29017311210423.

P8 = (a^3 b-a b^3+a^3 c-2 a^2 b c-2 a b^2 c-b^3 c-2 a b c^2-2 b^2 c^2-a c^3-b c^3) (a^4 b-a^3 b^2-a^2 b^3+a b^4+a^4 c+2 a^2 b^2 c+b^4 c+a^3 c^2+b^3 c^2-a^2 c^3-b^2 c^3-a c^4-b c^4) (a^4 b+a^3 b^2-a^2 b^3-a b^4+a^4 c-b^4 c-a^3 c^2+2 a^2 b c^2-b^3 c^2-a^2 c^3+b^2 c^3+a c^4+b c^4) : : , SEARCH = 5.43226360650240.

P9 = a (a b+b^2+2 a c+b c) (2 a b+a c+b c+c^2) (a^3 b+2 a^2 b^2+a b^3+a^3 c+2 a^2 b c+2 a b^2 c+b^3 c+2 a b c^2-a c^3-b c^3) (a^3 b-a b^3+a^3 c+2 a^2 b c+2 a b^2 c-b^3 c+2 a^2 c^2+2 a b c^2+a c^3+b c^3) (a^4 b+a^3 b^2-a^2 b^3-a b^4+a^4 c-b^4 c+a^3 c^2-2 a b^2 c^2+b^3 c^2-a^2 c^3+b^2 c^3-a c^4-b c^4) : : , SEARCH = -3.51184173102770.

P10 = (a^3 b-a b^3+a^3 c-2 a^2 b c-2 a b^2 c-b^3 c-2 a b c^2-2 b^2 c^2-a c^3-b c^3) (a^4 b+3 a^3 b^2+3 a^2 b^3+a b^4+a^4 c+4 a^3 b c+6 a^2 b^2 c+4 a b^3 c+b^4 c+a^3 c^2+6 a^2 b c^2+6 a b^2 c^2+3 b^3 c^2+a^2 c^3+4 a b c^3+3 b^2 c^3+a c^4+b c^4) (a^4 b+a^3 b^2+a^2 b^3+a b^4+a^4 c+4 a^3 b c+6 a^2 b^2 c+4 a b^3 c+b^4 c+3 a^3 c^2+6 a^2 b c^2+6 a b^2 c^2+3 b^3 c^2+3 a^2 c^3+4 a b c^3+3 b^2 c^3+a c^4+b c^4) : : , SEARCH = 4.22934202559219.

T1 = (a^4-2 a^2 b^2+b^4+2 a^2 b c+2 a b^2 c-2 a b c^2-c^4) (a^4-b^4+2 a^2 b c-2 a b^2 c-2 a^2 c^2+2 a b c^2+c^4) (a^10-3 a^8 b^2+2 a^6 b^4+2 a^4 b^6-3 a^2 b^8+b^10-2 a^8 b c-4 a^7 b^2 c-4 a^6 b^3 c+4 a^5 b^4 c+8 a^4 b^5 c-4 a^3 b^6 c-4 a^2 b^7 c+4 a b^8 c+2 b^9 c-3 a^8 c^2-4 a^7 b c^2-8 a^6 b^2 c^2+4 a^5 b^3 c^2+2 a^4 b^4 c^2-4 a^3 b^5 c^2+8 a^2 b^6 c^2+4 a b^7 c^2+b^8 c^2-4 a^6 b c^3+4 a^5 b^2 c^3-8 a^4 b^3 c^3+8 a^3 b^4 c^3+4 a^2 b^5 c^3-4 a b^6 c^3+2 a^6 c^4+4 a^5 b c^4+2 a^4 b^2 c^4+8 a^3 b^3 c^4-10 a^2 b^4 c^4-4 a b^5 c^4-2 b^6 c^4+8 a^4 b c^5-4 a^3 b^2 c^5+4 a^2 b^3 c^5-4 a b^4 c^5-4 b^5 c^5+2 a^4 c^6-4 a^3 b c^6+8 a^2 b^2 c^6-4 a b^3 c^6-2 b^4 c^6-4 a^2 b c^7+4 a b^2 c^7-3 a^2 c^8+4 a b c^8+b^2 c^8+2 b c^9+c^10) : : , SEARCH = 2.28317113324989.

T2 = a^2 (a^4 b+2 a^3 b^2+3 a^2 b^3+2 a b^4+a^4 c+3 a^3 b c+6 a^2 b^2 c+4 a b^3 c+2 b^4 c+3 a^3 c^2+4 a^2 b c^2+6 a b^2 c^2+3 b^3 c^2+3 a^2 c^3+3 a b c^3+2 b^2 c^3+a c^4+b c^4) (a^4 b+3 a^3 b^2+3 a^2 b^3+a b^4+a^4 c+3 a^3 b c+4 a^2 b^2 c+3 a b^3 c+b^4 c+2 a^3 c^2+6 a^2 b c^2+6 a b^2 c^2+2 b^3 c^2+3 a^2 c^3+4 a b c^3+3 b^2 c^3+2 a c^4+2 b c^4) (a^4 b^2+a^3 b^3-a^2 b^4-a b^5+2 a^4 b c+a^3 b^2 c-a^2 b^3 c-a b^4 c-b^5 c+a^4 c^2+a^3 b c^2+2 a^2 b^2 c^2-2 a b^3 c^2-2 b^4 c^2+a^3 c^3-a^2 b c^3-2 a b^2 c^3-2 b^3 c^3-a^2 c^4-a b c^4-2 b^2 c^4-a c^5-b c^5) : : , SEARCH = 6.79585517080318.

T3 = a^2 (a^4 b^2+a^3 b^3-a^2 b^4-a b^5+2 a^4 b c+a^3 b^2 c-a^2 b^3 c-a b^4 c-b^5 c+a^4 c^2+a^3 b c^2+2 a^2 b^2 c^2-2 a b^3 c^2-2 b^4 c^2+a^3 c^3-a^2 b c^3-2 a b^2 c^3-2 b^3 c^3-a^2 c^4-a b c^4-2 b^2 c^4-a c^5-b c^5) (a^6 b+a^5 b^2-2 a^4 b^3-2 a^3 b^4+a^2 b^5+a b^6+a^6 c-a^5 b c-a^4 b^2 c+2 a^3 b^3 c-a^2 b^4 c-a b^5 c+b^6 c+3 a^4 b c^2-3 a^3 b^2 c^2-3 a^2 b^3 c^2+3 a b^4 c^2+a^3 b c^3+6 a^2 b^2 c^3+a b^3 c^3+2 a^3 c^4-2 a^2 b c^4-2 a b^2 c^4+2 b^3 c^4-a^2 c^5-b^2 c^5-2 a c^6-2 b c^6) (a^6 b+2 a^3 b^4-a^2 b^5-2 a b^6+a^6 c-a^5 b c+3 a^4 b^2 c+a^3 b^3 c-2 a^2 b^4 c-2 b^6 c+a^5 c^2-a^4 b c^2-3 a^3 b^2 c^2+6 a^2 b^3 c^2-2 a b^4 c^2-b^5 c^2-2 a^4 c^3+2 a^3 b c^3-3 a^2 b^2 c^3+a b^3 c^3+2 b^4 c^3-2 a^3 c^4-a^2 b c^4+3 a b^2 c^4+a^2 c^5-a b c^5+a c^6+b c^6) : : , SEARCH = 3.74031047150095.

T4 = (a^3+2 a^2 b+a b^2+2 a^2 c+a b c+b^2 c+a c^2+b c^2) (a^4 b+2 a^3 b^2+2 a^2 b^3+2 a b^4+b^5+a^4 c+3 a^3 b c+5 a^2 b^2 c+5 a b^3 c+2 b^4 c+3 a^3 c^2+6 a^2 b c^2+5 a b^2 c^2+2 b^3 c^2+3 a^2 c^3+3 a b c^3+2 b^2 c^3+a c^4+b c^4) (a^4 b+3 a^3 b^2+3 a^2 b^3+a b^4+a^4 c+3 a^3 b c+6 a^2 b^2 c+3 a b^3 c+b^4 c+2 a^3 c^2+5 a^2 b c^2+5 a b^2 c^2+2 b^3 c^2+2 a^2 c^3+5 a b c^3+2 b^2 c^3+2 a c^4+2 b c^4+c^5) : : , SEARCH = 2.24679887970365.

T5 = (a^2+b^2+2 b c+c^2) (a^6+2 a^5 b+3 a^4 b^2+4 a^3 b^3+3 a^2 b^4+2 a b^5+b^6+2 a^5 c+4 a^4 b c+2 a^3 b^2 c+2 a^2 b^3 c+4 a b^4 c+2 b^5 c+a^4 c^2+2 a^3 b c^2+2 a^2 b^2 c^2+2 a b^3 c^2+b^4 c^2-2 a^2 b c^3-2 a b^2 c^3-a^2 c^4-4 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-a^2 b^4-2 a b^5-b^6+2 a^5 c+4 a^4 b c+2 a^3 b^2 c-2 a^2 b^3 c-4 a b^4 c-2 b^5 c+3 a^4 c^2+2 a^3 b c^2+2 a^2 b^2 c^2-2 a b^3 c^2-b^4 c^2+4 a^3 c^3+2 a^2 b c^3+2 a b^2 c^3+3 a^2 c^4+4 a b c^4+b^2 c^4+2 a c^5+2 b c^5+c^6) : : , SEARCH = 58.8751035274266.

T6 = (a-b-c) (a^4-2 a^2 b^2+b^4+2 a^2 b c+2 a b^2 c-2 a b c^2-c^4) (a^4-b^4+2 a^2 b c-2 a b^2 c-2 a^2 c^2+2 a b c^2+c^4) : : , SEARCH = 5.03818048975889.

These points are X(34262) -> X(34277) in ETC.

***

Triads of collinear points on K1135 :

Note that a line passing through X(941) meets K1135 again at two points which lie on a same rectangular circum-hyperbola. For instance, X2 and gX5019 lie on the Kiepert hyperbola.

The points Ti are the tangentials of other points Pi on K1135. See table below. Obviously, three collinear points have also collinear tangentials. See for example X4, X573, X2345 and their respective tangentials T1, T3, T4.

X2, X4, X11337

X2, X6, X5739

X2, X8, X386

X2, X573, P1

X2, X941, gX5019

X2, X959, gX4185

X2, P5, T6

X2, P6, P8

X4, X6, P5

X4, X8, X5739

X4, X386, P6

X4, X573, X2345

X4, X941, P2

X4, X959, P7

X4, gX4185, gX5019

X6, X8, X2345

X6, X386, X573

X6, X941, gX4185

X6, X959, P2

X6, P1, P6

X6, T1, T6

X8, X573, P8

X8, X959, gX5019

X8, X11337, T6

X8, P6, T3

X386, X941, P3

X386, X959, P9

X386, X2345, P10

X386, X5739, T4

X386, gX5019, P4

X386, P5, T5

X573, X941, X959

X573, X5739, T6

X941, X2345, P4

X941, P1, P9

X941, P7, T6

X959, X2345, P3

X959, X5739, P4

X2345, X11337, T5

X2345, P1, T2

X2345, P6, T6

X5739, P1, P10

X5739, P8, T2

X11337, P1, T4

X11337, P8, P10

X11337, T2, T3

gX4185, P1, P4

gX4185, P3, P8

gX4185, P9, T3

gX5019, P1, P3

gX5019, P2, T6

gX5019, P8, P9

P1, T1, T5

P2, P3, T3

P2, P4, P8

P4, P7, T3

P5, P8, T4

P5, P10, T3

T1, T3, T4