X3751

X1721com[INVERSE(n(HEXYL TRIANGLE))]

Let A'B'C' denote the inverse of the normalized hexyl triangle. Then A', expressed in homogeneous barycentrics (i.e., not normalized), is given by
< 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) { 1, 6 }: Incenter–Symmedian Line { 10, 69 }: Central Line L(10, 69) { 43, 57 }: Central Line L(43, 57) { 42, 63 }: Central Line L(42, 63) { 8, 193 }: Central Line L(8, 193) 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_6 = SYMMEDIAN POINT (LEMOINE POINT, GREBE POINT) (0.6000, 0.2672) X_10 = SPIEKER CENTER (0.5204, 0.1833) X_69 = SYMMEDIAN POINT OF THE ANTICOMPLEMENTARY TRIANGLE (0.4122, 0.0654) X_43 = -CEVA CONJUGATE OF X(1) (0.6358, 0.2963) X_57 = ISOGONAL CONJUGATE OF X(9) (0.6138, 0.3110) X_42 = CROSSPOINT OF INCENTER AND SYMMEDIAN POINT (0.5880, 0.2495) X_63 = ISOGONAL CONJUGATE OF X(19) (0.4566, 0.0823) X_8 = NAGEL POINT (0.4694, 0.1333) X_193 = -CEVA CONJUGATE OF X(2) (0.7878, 0.4690) X_3751 = com[INVERSE(n(HEXYL TRIANGLE))] (0.6286, 0.3011) A B C X1 X6 X10 X69 X43 X57 X42 X63 X8 X193 X3751 (a, b, c) = (5, 6, 7) | AB = 1

X3751 (X1721com[INVERSE(n(HEXYL TRIANGLE))]) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

Trilinears: \(aS - rS_{\omega } : bS - rS_{\omega } : cS - rS_{\omega }\)
Barycentrics: \(a(a^{2} - b^{2} - c^{2} + 2ab + 2ac) : :\)
Search Value (6, 9, 13): \(0.295508480674\)
▸ Linear Combinations1 combination
▸ Incident Central Lines51 lines

{1, 6}, {7, 1738}, {8, 193}, {10, 69}, {31, 1331}, {33, 1957}, {40, 511}, {42, 63}, {43, 57}, {46, 3293}, {55, 1707}, {58, 1792}, {78, 1468}, {81, 612}, {141, 1698}, {165, 1350}, {171, 200}, {182, 3576}, {210, 940}, {240, 2331}, {314, 989}, {355, 3564}, {386, 988}, {517, 1351}, {519, 1992}, {524, 3416}, {614, 3315}, {651, 2263}, {872, 2274}, {896, 2177}, {899, 3306}, {942, 1722}, {952, 1353}, {968, 3219}, {971, 1721}, {978, 3333}, {982, 2999}, {990, 1814}, {1125, 3618}, {1420, 1428}, {1445, 1458}, {1697, 3056}, {2330, 3601}, {2647, 3340}, {2648, 3577}, {2854, 2948}, {3094, 3097}, {3158, 3550}, {3216, 3338}, {3218, 3240}, {3589, 3624}

▸ Functional Relations2 relations

Notes

2a(b + c) : \(a^{2}\) + \(b^{2}\) - \(c^{2}\) : \(a^{2}\) - \(b^{2}\) + \(c^{2}\)

In general, if a triangle DEF is perspective to a given triangle GHI from a point P and also perspector to another triangle JKL from a point Q, then, clearly, lines joining P to G, H, I, and line joining Q to J,K,L concur in pairs to form DEF. Thus, if constructions are known for G,H,I,P,J,K,I,Q, then a construction for DEF easily follows. In particular, the triangle A'B'C' can be constructed in several ways, as it is perspective to the following triangles with perspectors:

ABC from X4, orthic triangle from X4, medial triangle from X2, intouch triangle from X65, extouch triangle from X10, Euler triangle X4