X3640

1st KIRIKAMI POINT

j(a,b,c) = \(a^{4}\) + \(b^{4}\) + \(c^{4}\) - \(8b^{3}\)c - \(8bc^{3}\) + \(6b^{2}\)\(c^{2}\) - \(2a^{2}\)\(b^{2}\) - \(2a^{2}\)\(c^{2}\) + 8abc(b+c-a).
< A B C >: Reference Triangle △ABC [ A B ]: Sideline AB (Base c = 1) [ B C ]: Sideline BC (Side a) [ C A ]: Sideline CA (Side b) { 1, 6 }: Incenter–Symmedian Line { 8, 175 }: Central Line L(8, 175) { 517, 1160 }: Central Line L(517, 1160) { 481, 2550 }: Central Line L(481, 2550) { 3084, 3681 }: Central Line L(3084, 3681) { 3083, 3873 }: Central Line L(3083, 3873) A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6122, 0.5999) X_1 = INCENTER (0.5714, 0.2333) X_6 = SYMMEDIAN POINT (LEMOINE POINT, GREBE POINT) (0.6000, 0.2672) X_8 = NAGEL POINT (0.4694, 0.1333) X_175 = ISOPERIMETRIC POINT (0.4271, -0.1006) X_517 = ISOGONAL CONJUGATE OF X(104) (0.8180, 0.7154) X_1160 = CIRCUMCENTER OF OUTER GREBE TRIANGLE (0.1258, -0.5160) X_481 = 1st EPPSTEIN POINT (0.6148, 0.3335) X_2550 = INSIMILICENTER(SPIEKER CIRCLE, (X(4),2R)) (0.5298, 0.2051) X_3084 = INTERSECTION OF LINES X(1)X(2) AND X(37)X(493) (0.3591, 0.0252) X_3681 = com(EXTOUCH TRIANGLE) (0.4476, 0.0933) X_3083 = INTERSECTION OF LINES X(1)X(2) AND X(37)X(494) (0.5529, 0.2152) X_3873 = com[INVERSE(n(3rd EULER TRIANGLE))] (0.6272, 0.3066) X_3640 = 1st KIRIKAMI POINT (0.4645, 0.1063) A B C X1 X6 X8 X175 X517 X1160 X481 X2550 X3084 X3681 X3083 X3873 X3640 (a, b, c) = (5, 6, 7) | AB = 1

X3640 (1st KIRIKAMI POINT) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

Barycentrics: \(a (b^{2} + c^{2} - a (b + c) + S) : :\)
Search Value (6, 9, 13): \(5.904594223743\)
▸ Linear Combinations1 combination

  • X3640 = \(3\) X1 - \(2\) X45714, \(3\) X3576 - \(4\) X45715, X3641 - \(4\) X45713, \(3\) X3641 - \(4\) X45714, \(3\) X5860 - X12628, \(3\) X45713 - X45714

▸ Incident Central Lines70 lines

{1, 6}, {3, 11498}, {8, 175}, {10, 5590}, {30, 16131}, {40, 11825}, {55, 8199}, {65, 10976}, {142, 30342}, {144, 30334}, {145, 30333}, {200, 13388}, {355, 6214}, {481, 2550}, {515, 5870}, {517, 1160}, {519, 5860}, {537, 24832}, {730, 6272}, {944, 10784}, {946, 6201}, {952, 5874}, {976, 36567}, {982, 8945}, {1374, 5880}, {1385, 26348}, {1482, 11917}, {1703, 8416}, {1829, 11389}, {1837, 10926}, {2800, 12754}, {2802, 13270}, {2809, 6213}, {3057, 10928}, {3083, 3873}, {3084, 3681}, {3244, 26340}, {3576, 45553}, {3579, 35247}, {4847, 13390}, {5252, 10924}, {5393, 25568}, {5405, 24477}, {5587, 10515}, {5594, 9798}, {5657, 10518}, {5689, 10513}, {5698, 31567}, {5881, 6278}, {6001, 6257}, {6203, 30320}, {8218, 12440}, {8219, 12441}, {8975, 8983}, {9941, 9995}, {10793, 12194}, {11902, 12438}, {12697, 40268}, {13950, 13971}, {16232, 39959}, {18480, 18511}, {18525, 26346}, {26338, 45717}, {26344, 45711}, {26345, 45712}, {26347, 45718}, {35641, 35794}, {35642, 35793}, {36480, 36536}, {45551, 45716}

▸ Functional Relations3 relations

  • X3640 = reflection of Xi in Xj for these {i,j}: {1, 45713}, {3641, 1}, {12698, 1160}, {45720, 3244}

  • X3640 = X1-of-outer-Grebe-triangle

  • X3640 = {Xi,Xj}-harmonic conjugate of Xk for these (i,j,k): {1, 6, 11371}, {1, 3751, 18991}, {1, 5223, 30557}, {1, 5588, 6}, {1, 5589, 7969}, {1, 18992, 11370}, {1, 19003, 1386}, {1, 19004, 45398}, {1, 45426, 18992}, {6, 5604, 1}, {6, 10930, 10041}, {6, 10932, 10049}, {8, 1270, 5688}, {3242, 7968, 1}, {4663, 45398, 19004}, {5588, 5604, 11371}, {5605, 44636, 1}, {6762, 7174, 3641}, {7969, 45476, 1}, {8199, 8206, 55}, {10920, 10922, 6214}, {11498, 22757, 3}, {12594, 19049, 10040}, {12595, 19047, 10048}, {26349, 26350, 6}, {45422, 45729, 3301}, {45424, 45728, 3299}

Notes

Seiichi Kirikami (November 1, 2010), defined a point P having actual trilinear distances (p,q,r) by the equations a + p = b + q = c + r and a point P' with distances (p’,q’,r’) satisfying p - a = q - b = r - c. Here, P = X3640 and P' = X3641. Kirikami also discovered a bicentric pair of points satisfying p + b = q + c = r + a and another bicentric pair satisfying p - b = q - c = r - a. The six point are given by homogeneous trilinears \(t(a,b,c) : t(b,c,a) : t(c,a,b)\) as follows:

X3640: t(a,b,c) = \(b^{2}\) + \(c^{2}\) - ab - ac + 2*area(ABC)

X3641: t(a,b,c) = \(b^{2}\) + \(c^{2}\) - ab - ac - 2*area(ABC)

K1: t(a,b,c) = ac - \(b^{2}\) + 2*area(ABC)

K2: t(a,b,c) = ab - \(c^{2}\) + 2*area(ABC)

K3: t(a,b,c) = ac - \(b^{2}\) + 2*area(ABC)

K4: t(a,b,c) = ab -\(cb^{2}\) + 2*area(ABC)

Peter Moses (November 1, 2010) found that the six points lie on an ellipse, E, here called the Kirikami-Moses ellipse. The center of E is the incenter of ABC, and an equation for E is as follows:

e(a,b,c,x,y,z) + e(b,c,a,y,z,x) + e(c,a,b,z,x,y) = 0, where

e(a,b,c,x,y,z) = h(a,b,c)\(x^{2}\) - j(a,b,c)yz, where

\(h(a,b,c) = (5a^{4} + b^{4} + c^{4} - 4a^{3}b - 4a^{3}c - 4a^{2}bc + 2b^{2}c^{2} + 2c^{2}a^{2} + 2a^{2}b^{2})\)

and

Moses found six more points on E:

M1 = (b - c) (a + b + c) + 2*\(\sqrt{3}\)*area(ABC) : :

  • M2 = (b - c) (a + b + c) - 2*\(\sqrt{3}\)*area(ABC) : :

  • M3 = 2ab + ac - \(b^{2}\) - \(2c^{2}\) + 2*\(\sqrt{3}\)*area(ABC) : :

  • M4 = 2ac + ab - \(c^{2}\) - \(2b^{2}\) + 2*\(\sqrt{3}\)*area(ABC) : :

  • M5 = 2ab + ac - \(b^{2}\) - \(2c^{2}\) - 2*\(\sqrt{3}\)*area(ABC) : :

  • M6 = 2ac + ab - \(c^{2}\) - \(2b^{2}\) - 2*\(\sqrt{3}\)*area(ABC) : : ‘

These six points comprise three bicentric pairs: {M1, M2}, {M3, M4}, {M5, M6}.

Further properties found by Moses:

  1. The incenter is the midpoint of the segments X3640X3641, K1K2, and K3K4.

  2. X3640 and X3641 lie on the line X1X6.

  3. The lines K1K4 and K2K3 are parallel to the line X1X6.

  4. The lines X3640K4, K1K3, and X3641K2 are parallel; indeed, the reflection of the first in the second is the third.

  5. The lines X3640K3, K2K3, and X3641K1 are parallel, and the reflection of the first in the second is the third.

  6. The lines tangent to E at X3640 and X3641 are perpendicular to the line X1X3, as are the lines K1K2 and K3K4.

  7. The following four lines are parallel: the tangents to E at K2 and K4, and X3640K1 and X3641K3.

  8. The following four lines are parallel: the tangents to E at K1 and K3, and X3640K2 and X3641K4.