HYACINTHOS 28929

[Antreas P. Hatzipolakis]:

Let ABC be a triangle and L a line.

Three parallels to L through A, B, C intersect the curcumcircle again at A1, B1, C1, resp.

Let A2, B2, C2 be the reflections of A1, B1, C1 in BC, CA, AB, rersp.

The NPCs of ABC and A2B2C2 are tangent,

Reference: Ioannis Panakis (in a 1961 paper in Greek)

Which is the point of tangency?
Special L = Euler line, Brocard axis, OI line


[Peter Moses]:

Hi Antreas,

For some line L = L1 x + L2 y + L3 z = 0, the tangency point is
(b^2 (L2 - L1) - c^2 (L3 - L1)) (L2 - L3) : :

----------------------------
OI line -> X(119)
Euler line -> X(113)
Brocard axis -> X(114)
GK line -> X(126)
Orthic axis -> X(125)
anti-Orthic axis -> X(11)
Lemoine axis -> X(115)
De Longchamps axis -> X(125)
Gergonne line -> X(116)
Soddy line -> X(118)
Nagel line -> X(121)
Fermat line -> X(16188)
van Aubel line -> X(132)
IK line -> X(120)
IH line -> X(117)
----------------------------

IN line ->  COMPLEMENT OF X(953)

 

= (2 a^4-2 a^3 b-a^2 b^2+2 a b^3-b^4-2 a^3 c+4 a^2 b c-2 a b^2 c-a^2 c^2-2 a b c^2+2 b^2 c^2+2 a c^3-c^4) (a^4 b^2-2 a^2 b^4+b^6-2 a^3 b^2 c+2 a^2 b^3 c+2 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+4 b^3 c^3-2 a^2 c^4+2 a b c^4-b^2 c^4-2 b c^5+c^6) : : 

 

= lies on the nine-point circle and these lines: {2, 953}, {3, 2222}, {4, 901}, {5, 3259}, {11, 517}, {12, 3025}, {115, 2245}, {119, 513}, {121, 20316}, {123, 5123}, {124, 3814}, {136, 860}, {952, 6073}, {1210, 24201}, {1482, 5516}, {1772, 24639}, {1878, 5521}, {5190, 8756}, {5520, 9956}, {7649, 20619}, {7741, 23153}, {15611, 17734}, {15612, 26446}, {17606, 23152}

= complement of X(953).
= reflection of X(i) and X(j) for these {i,j}: {3, 22102}, {3259, 5}
= complement of the isogonal of X(952)
= X(i)-complementary conjugate of X(j) for these (i,j): {1, 952}, {952, 10}, {2265, 2}
= X(4)-Ceva conjugate of X(952)
----------------------------

NK line ->  COMPLEMENT OF X(3563)

 

= (a^2-b^2-c^2) (2 a^4-a^2 b^2+b^4-a^2 c^2-2 b^2 c^2+c^4) (a^6 b^2-a^4 b^4-a^2 b^6+b^8+a^6 c^2-4 a^4 b^2 c^2+3 a^2 b^4 c^2-4 b^6 c^2-a^4 c^4+3 a^2 b^2 c^4+6 b^4 c^4-a^2 c^6-4 b^2 c^6+c^8) : : 

 

= lies on the nine-point circle and these lines: {2, 136}, {3, 115}, {4, 3565}, {5, 5139}, {22, 15241}, {114, 2974}, {122, 30771}, {125, 343}, {127, 11585}, {135, 427}, {137, 6676}, {468, 16178}, {858, 16221}, {1560, 2967}, {2072, 5099}, {2453, 6721}, {3258, 5159}, {3546, 28438}, {3549, 14669}, {5512, 15760}, {5522, 7386}, {10691, 11792}, {14672, 18531}, {16051, 30789}

= complement of X(3563)
= midpoint of X(4) and X(3565)
= Inverse of X(925) in the orthoptic circle of the Steiner inellipe
= complement of the isogonal of X(3564)
= X(i)-complementary conjugate of X(j) for these (i,j): {1, 3564}, {230, 226}, {293, 6036}, {460, 24005}, {656, 868}, {1692, 16583}, {1733, 5}, {3564, 10}, {4226, 8062}, {4575, 6132}, {8772, 6}, {17462, 15595}
= X(4)-Ceva conjugate of X(3564)
----------------------------

 

X(6) X(17) X(18) line -> X(2)X(137)∩X(115)X(140)

 

= (2 a^6-4 a^4 b^2+3 a^2 b^4-b^6-4 a^4 c^2+b^4 c^2+3 a^2 c^4+b^2 c^4-c^6) (a^6 b^2-a^4 b^4-a^2 b^6+b^8+a^6 c^2-6 a^4 b^2 c^2+4 a^2 b^4 c^2-5 b^6 c^2-a^4 c^4+4 a^2 b^2 c^4+8 b^4 c^4-a^2 c^6-5 b^2 c^6+c^8) : : 

 

= lies on the nine-point circle and these lines: {2, 137}, {115, 140}, {125, 3819}, {1368, 20625}, {1594, 5139}, {3258, 30745}

= Inverse of X(930) in the orthoptic circle of the Steiner inellipe
= complement of the isogonal of X(5965)
= X(i)-complementary conjugate of X(j) for these (i,j): {1, 5965}, {5965, 10}
= X(4)-Ceva conjugate of X(5965)
----------------------------

X(2) X(7) -> COMPLEMENT OF X(2291)

 

= (2 a^2-a b-b^2-a c+2 b c-c^2) (a^2 b^2-2 a b^3+b^4+a b^2 c+b^3 c+a^2 c^2+a b c^2-4 b^2 c^2-2 a c^3+b c^3+c^4) : : 

 

= lies on the nine-point circle and these lines: {2, 2291}, {4, 28291}, {11, 142}, {115, 15903}, {116, 2886}, {120, 4928}, {123, 18589}, {124, 141}, {125, 17052}, {1125, 15746}, {1566, 16593}, {3259, 20335}, {3452, 5514}, {3817, 5511}, {4728, 5513}, {5074, 5087}, {5274, 26140}, {5510, 24220}, {5518, 20528}, {5886, 25642}, {25532, 26105}

= complement of X(2291)
= reflection of X(15746) and X(1125)
= Inverse of X(9086) in the orthoptic circle of the Steiner inellipe.
= complement of the isogonal of X(527).
= X(i)-complementary conjugate of X(j) for these (i,j): {1, 527}, {2, 5087}, {9, 5199}, {527, 10}, {651, 6366}, {1055, 37}, {1155, 2}, {1323, 142}, {1638, 11}, {6174, 16594}, {6366, 26932}, {6510, 3}, {6603, 9}, {6610, 1}, {6745, 3452}, {14392, 13609}, {14413, 1086}, {14414, 16596}, {15730, 6594}, {23346, 905}, {23710, 226}, {23890, 522}, {24685, 17793}, {30574, 8287}, {30806, 141}
= X(4)-Ceva conjugate of X(527)
---------------------------------------------------

X(1) X(21) -> COMPLEMENT OF X(759)

 

= (b+c)^2 (-a^2+b^2-b c+c^2) (a^3+b^3-a b c-b^2 c-b c^2+c^3) : : 

 

= lies on the nine-point circle and these lines: {2, 759}, {4, 6011}, {5, 25652}, {10, 125}, {11, 214}, {12, 1365}, {37, 115}, {116, 3739}, {123, 21530}, {124, 960}, {127, 18589}, {338, 15065}, {429, 5521}, {860, 13999}, {1211, 15614}, {1283, 5051}, {1511, 5499}, {1656, 14663}, {1698, 21381}, {2679, 19563}, {2802, 8286}, {3259, 11813}, {3634, 5993}, {3814, 5520}, {3936, 4867}, {4197, 19642}, {5074, 20529}, {5099, 16597}, {5497, 30117}, {5510, 9955}, {5511, 30444}, {5515, 16586}, {6702, 8287}, {8728, 25448}, {21232, 21253}, {24220, 30448}, {26364, 27686}

= 5 X[1698] - X[21381]

= complement of X(759).
= inverse of X(3031) in the Spieker radical circle.
= inverse of X(9070) in the orthoptic circle of the Steiner inellipe.
= complement of the isogonal of X(758).
= X(i)-complementary conjugate of X(j) for these (i,j): {1, 758}, {10, 3814}, {36, 1125}, {37, 908}, {42, 44}, {55, 7359}, {65, 1737}, {109, 21180}, {110, 6370}, {320, 3741}, {321, 21237}, {526, 6741}, {654, 4858}, {758, 10}, {860, 5}, {1443, 3742}, {1464, 1}, {1835, 1210}, {1870, 942}, {1983, 14838}, {2245, 2}, {2323, 5745}, {2594, 6149}, {2610, 8287}, {3028, 6739}, {3218, 3739}, {3724, 37}, {3936, 141}, {3960, 17761}, {4053, 1211}, {4242, 8062}, {4282, 16579}, {4511, 960}, {4551, 3738}, {4557, 1639}, {4559, 10015}, {4585, 4369}, {4674, 6702}, {4707, 116}, {6370, 125}, {6739, 113}, {6742, 526}, {7113, 3666}, {8818, 3580}, {17078, 17050}, {18593, 142}, {20924, 21240}, {21828, 1086}, {23493, 21331}
= X(i)-Ceva conjugate of X(j) for these (i,j): {4, 758}, {3952, 6370}
---------------------------------------------------

Best regards,
Peter Moses.