X1480

1st SHADOW POINT

X1480 = X1-Ceva conjugate of X999
< 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) { 6, 517 }: Central Line L(6, 517) { 3, 902 }: Central Line L(3, 902) { 1, 1406 }: Central Line L(1, 1406) { 651, 1000 }: Central Line L(651, 1000) 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_999 = MIDPOINT OF X(1) AND X(57) (0.5926, 0.2722) X_6 = SYMMEDIAN POINT (LEMOINE POINT, GREBE POINT) (0.6000, 0.2672) X_517 = ISOGONAL CONJUGATE OF X(104) (0.8180, 0.7154) X_3 = CIRCUMCENTER (0.5000, 0.1021) X_902 = CROSSPOINT OF X(6) AND X(106) (1.2500, 1.2247) X_1406 = -BETH CONJUGATE OF X(46) (0.6653, 0.5088) X_651 = TRILINEAR POLE OF LINE X(1)X(3) (0.3048, 0.5599) X_1000 = ISOGONAL CONJUGATE OF X(999) (0.5463, 0.1961) X_1480 = 1st SHADOW POINT (0.5545, 0.1837) A B C X1 X999 X6 X517 X3 X902 X1406 X651 X1000 X1480 (a, b, c) = (5, 6, 7) | AB = 1

X1480 (1st SHADOW POINT) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

Trilinears: \(f(A,B,C) : f(B,C,A) : f(C,A,B)\), where \(f(A,B,C) = (2 - \cos A)(2 + \cos A - \cos B - \cos C)\)
Barycentrics: \((\sin A)f(A,B,C) : (\sin B)f(B,C,A) : (\sin C)f(C,A,B)\)
Search Value (6, 9, 13): \(3.671196840286\)
▸ Linear Combinations1 combination

  • Antreas P. Hatzipolakis and Paul Yiu, Pedal Triangles and Their Shadows, :cite:`Forum Geometricorum <forum-geometricorum>` 1 (2001) 81-90. ( X1480is point M on page 88. See X6580for the 2nd Shadow Point.

▸ Incident Central Lines4 lines

{1, 1406}, {3, 902}, {6, 517}, {651, 1000}

▸ Functional Relations1 relation