X178

2nd MID-ARC POINT

X178 = complement of X188
< A B C >: Reference Triangle △ABC < M_a M_b M_c >: Medial Triangle △M_a M_b M_c [ A B ]: Sideline AB (Base c = 1) [ B C ]: Sideline BC (Side a) [ C A ]: Sideline CA (Side b) { 2, 188 }: Central Line L(2, 188) { 8, 236 }: Central Line L(8, 236) A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6122, 0.5999) M_a = Midpoint M_a of BC (0.8061, 0.2999) M_b = Midpoint M_b of CA (0.3061, 0.2999) M_c = Midpoint M_c of AB (0.5000, 0.0000) X_2 = CENTROID (0.5374, 0.2000) X_188 = 2nd MID-ARC POINT OF ANTICOMPLEMENTARY TRIANGLE (0.5245, 0.1802) X_8 = NAGEL POINT (0.4694, 0.1333) X_236 = -CEVA CONJUGATE OF X(188) (0.5520, 0.2182) X_178 = 2nd MID-ARC POINT (0.5439, 0.2098) A B C Ma Mb Mc X2 X188 X8 X236 X178 (a, b, c) = (5, 6, 7) | AB = 1

X178 (2nd MID-ARC POINT) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

Trilinears: \((\cos B/2 + \cos C/2) \csc A : (\cos C/2 + \cos A/2) \csc B : (\cos A/2 + \cos B/2) \csc C\)
Barycentrics: \(\cos B/2 + \cos C/2 : \cos C/2 + \cos A/2 : \cos A/2 + \cos B/2\)
Search Value (6, 9, 13): \(2.359244703059\)
▸ Incident Central Lines2 lines

{2, 188}, {8, 236}

▸ Functional Relations2 relations

Notes

Let A',B',C' be the first points of intersection of the angle bisectors of ABC with its incircle. Let A″,B″,C″ be the midpoints of segments BC,CA,AB, respectively. The lines A'A″,B'B″,C'C″ concur in X178.

Clark Kimberling, Problem 804, Nieuw Archikef voor Wiskunde 6 (1988) 170.