X1285

LEMOINE HOMOTHETIC CENTER

X1285 is the homothetic center of the antipedal triangle of X2 and the pedal triangle of X6. (Darij Grinberg, Hyacinthos #6577, 2/21/03). See also X3066.
< 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) { 4, 32 }: Central Line L(4, 32) { 6, 376 }: Central Line L(6, 376) { 497, 609 }: Central Line L(497, 609) { 193, 1003 }: Central Line L(193, 1003) { 172, 1058 }: Central Line L(172, 1058) { 2, 1384 }: Central Line L(2, 1384) A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6122, 0.5999) X_4 = ORTHOCENTER (0.6122, 0.3958) X_32 = 3rd POWER POINT (0.6400, 0.3332) X_6 = SYMMEDIAN POINT (LEMOINE POINT, GREBE POINT) (0.6000, 0.2672) X_376 = CENTROID OF THE ANTIPEDAL TRIANGLE OF X(2) (0.4626, 0.0042) X_497 = CROSSPOINT OF GERGONNE POINT AND NAGEL POINT (0.5593, 0.1851) X_609 = INTERSECTION OF LINES X(1)X(32) AND X(6)X(36) (0.6191, 0.3028) X_193 = -CEVA CONJUGATE OF X(2) (0.7878, 0.4690) X_1003 = INTERCEPT OF EULER LINE AND LINE X(6)X(99) (0.5673, 0.2782) X_172 = TRILINEAR PRODUCT X(6)*X(171) (0.6080, 0.2866) X_1058 = POINT ALKALUROPS (0.5662, 0.2123) X_2 = CENTROID (0.5374, 0.2000) X_1384 = 2nd GRINBERG HOMOTHETIC CENTER (0.6750, 0.3910) X_1285 = LEMOINE HOMOTHETIC CENTER (0.6375, 0.3389) A B C X4 X32 X6 X376 X497 X609 X193 X1003 X172 X1058 X2 X1384 X1285 (a, b, c) = (5, 6, 7) | AB = 1

X1285 (LEMOINE HOMOTHETIC CENTER) 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 \(f(a,b,c) = bc(3a^{2} + b^{2} - c^{2})(3a^{2} - b^{2} + c^{2})\)
Barycentrics: \(af(a,b,c) : bf(b,c,a) : cf(c,a,b)\)
Search Value (6, 9, 13): \(0.102966916856\)
▸ Incident Central Lines9 lines

{2, 1384}, {4, 32}, {6, 376}, {99, 1992}, {172, 1058}, {193, 1003}, {497, 609}, {631, 3053}, {1056, 1914}

Notes

Let T denote the antipedal triangle of X2, and let T(m) denote X(m)-of-T. Then T(m) is a triangle center of the reference triangle ABC. The appearance of (m,n) in the following list means that T(m) = Xn. Note the special case m = 1285.

(2,376), (6,2), (115,99), (125,1296), (597,3), (599,20), (1285,1285), (1992,4), (2030,187), (2549,69)

Let U denote the pedal triangle of X6, and let U(m) denote X(m)-of-U. Then U(m) is a triangle center of the reference triangle ABC. The appearance of (m,n) in the following list means that U(m) = Xn. (This list and the one just above were contributed by Peter Moses, 11/15/2007.)

(2,6), (3,597), (4,1992), (20,599), (69,2549), (99,115), (187,2030), (376,2), (1285, 1285), (1296, 125)

Let A'B'C' be the antipedal triangle of X2. The centroid of A'B'C' is X376. Let A″B″C″ be the antipedal triangle, wrt A'B'C', of X376. A″B″C″ is homothetic to ABC, and the center of homothety is X1285. (Randy Hutson, March 21, 2019)

Let A'B'C' be the pedal triangle of X6. Let K' be X6 of A'B'C'. Let A″B″C″ be the pedal triangle, wrt A'B'C', of K'. A″B″C″ is homothetic to ABC, and the center of homothety is X1285. (Randy Hutson, March 21, 2019)