X495

JOHNSON MIDPOINT

X495 is the midpoint of segments C1-to-P1, C2-to-P2, C3-to-P3 in the Johnson four-circle configuration.
< 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, 956 }: Central Line L(2, 956) { 1, 5 }: Feuerbach Line { 30, 55 }: Central Line L(30, 55) { 10, 141 }: Central Line L(10, 141) { 56, 140 }: Central Line L(56, 140) 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_956 = INTERSECTION OF LINES X(1)X(6) AND X(3)X(8) (0.4790, 0.1235) X_1 = INCENTER (0.5714, 0.2333) X_5 = NINE-POINT CENTER (0.5561, 0.2489) X_30 = EULER INFINITY POINT (0.7469, 0.7481) X_55 = INSIMILICENTER(CIRCUMCIRCLE, INCIRCLE) (0.5490, 0.1921) X_10 = SPIEKER CENTER (0.5204, 0.1833) X_141 = COMPLEMENT OF SYMMEDIAN POINT (0.5061, 0.1663) X_56 = EXSIMILICENTER(CIRCUMCIRCLE, INCIRCLE) (0.6316, 0.3438) X_140 = MIDPOINT OF X(3) AND X(5) (0.5281, 0.1755) X_495 = JOHNSON MIDPOINT (0.5666, 0.2382) A B C Ma Mb Mc X2 X956 X1 X5 X30 X55 X10 X141 X56 X140 X495 (a, b, c) = (5, 6, 7) | AB = 1

X495 (JOHNSON MIDPOINT) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

Trilinears: \(2 + \cos(B - C) : 2 + \cos(C - A) : 2 + \cos(A - B)\)
Barycentrics: \((\sin A)[2 + \cos(B - C)] : (\sin B)[2 + \cos(C - A)] : (\sin C)[2 + \cos(A - B)]\)
Search Value (6, 9, 13): \(1.479183355337\)
▸ Linear Combinations1 combination
▸ Incident Central Lines18 lines

{1, 5}, {2, 956}, {3, 388}, {4, 390}, {8, 442}, {10, 141}, {30, 55}, {35, 550}, {36, 549}, {56, 140}, {202, 395}, {203, 396}, {226, 517}, {381, 497}, {392, 908}, {429, 1068}, {529, 993}, {612, 1060}

▸ Functional Relations2 relations

Notes

Roger A. Johnson, Advanced Euclidean Geometry, Dover, New York, 1960, page 75.

Peter Yff, “Three concurrent congruent circles ‘inscribed’ in a triangle,” manuscript, 1998; X495 is the point R on page 5. (See also X496-X499 and X1478, X1479.)