X3646

POINT ALMACH

Denote the incenter and excenters by I, IA, IB, IC. Let KA be the centroid of triangle BCI, and define KB and KC cyclically. The lines IAKA, IBKB, ICKC concur in X3646.
< 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) { 2, 40 }: Central Line L(2, 40) { 1, 210 }: Central Line L(1, 210) { 57, 191 }: Central Line L(57, 191) { 10, 1058 }: Central Line L(10, 1058) { 9, 1125 }: Central Line L(9, 1125) { 11, 1697 }: Central Line L(11, 1697) A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6122, 0.5999) X_2 = CENTROID (0.5374, 0.2000) X_40 = BEVAN POINT (0.4286, -0.0292) X_1 = INCENTER (0.5714, 0.2333) X_210 = -CEVA CONJUGATE OF X(37) (0.4925, 0.1466) X_57 = ISOGONAL CONJUGATE OF X(9) (0.6138, 0.3110) X_191 = -CEVA CONJUGATE OF X(1) (0.4859, 0.1209) X_10 = SPIEKER CENTER (0.5204, 0.1833) X_1058 = POINT ALKALUROPS (0.5662, 0.2123) X_9 = MITTENPUNKT (0.5110, 0.1615) X_1125 = COMPLEMENT OF X(10) (0.5459, 0.2083) X_11 = FEUERBACH POINT (0.7347, 0.0667) X_1697 = INSIMILICENTER(BEVAN CIRCLE, INCIRCLE) (0.5449, 0.1845) X_3646 = POINT ALMACH (0.5334, 0.1916) A B C X2 X40 X1 X210 X57 X191 X10 X1058 X9 X1125 X11 X1697 X3646 (a, b, c) = (5, 6, 7) | AB = 1

X3646 (POINT ALMACH) 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 \(f(A,B,C) = 5 + \cos A + (3 + \cos A)(\csc A)(\sin B + \sin C)\)
Search Value (6, 9, 13): \(2.801690292702\)
▸ Alternate Coordinates2 forms

  • Trilinears: \(1 + 2(1 + \cos^{2}(A/2)) \cos(B/2) \cos(C/2) \cos(A/2)\)

  • Trilinears: \(a^{3} + a^{2} (b + c) - a (b^{2} + 10 b c + c^{2}) - (b + c) (b^{2} + 6 b c + c^{2}) : :\)

▸ Linear Combinations1 combination
▸ Incident Central Lines11 lines

{1, 210}, {2, 40}, {3, 2951}, {9, 1125}, {10, 1058}, {11, 1697}, {57, 191}, {405, 1490}, {748, 1453}, {936, 1001}, {978, 1045}

Notes

Of the 2 intersections of the Bevan circle and line BC, let Ab be the one closer to B, and define Bc and Ca cyclically. Let Ac be the one closer to C, and define Ba and Cb cyclically. Let Ab' = {B,C}-harmonic conjugate of Ab, and define Bc' and Ca' cyclically. Let Ac' = {B,C}-harmonic conjugate of Ac, and define Ba' and Cb' cyclically. The points Ab', Ac', Bc', Ba', Ca', Cb' lie on an ellipse centered at X3646. (Randy Hutson, December 10, 2016)