X1981

VEGA TRANSFORM OF X647

lies on the line [\(x:y:z\)] that has x,y,z as coefficients. (The line [\(x:y:z\)] is the trilinear polar of the isogonal conjugate of X.) Thus, the Vega transform of X647 lies on the Euler line.
< 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, 3 }: Euler Line { 651, 653 }: Central Line L(651, 653) { 662, 811 }: Central Line L(662, 811) 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_3 = CIRCUMCENTER (0.5000, 0.1021) X_4 = ORTHOCENTER (0.6122, 0.3958) X_651 = TRILINEAR POLE OF LINE X(1)X(3) (0.3048, 0.5599) X_653 = TRILINEAR POLE OF LINE X(1)X(4) (0.5097, 0.6030) X_662 = TRILINEAR POLE OF LINE X(1)X(21) (0.1811, 0.4951) X_811 = ISOGONAL CONJUGATE OF X(810) (0.4151, 0.5616) X_1981 = VEGA TRANSFORM OF X(647) (0.7074, 0.6447) A B C X2 X3 X4 X651 X653 X662 X811 X1981 (a, b, c) = (5, 6, 7) | AB = 1

X1981 (VEGA TRANSFORM OF X647) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

Trilinears: \([\sin 2B \sin(C - A) - \sin 2C \sin(A - B)]/[\sin 2A \sin(B - C)] : :\)
Search Value (6, 9, 13): \(0.452286360966\)
▸ Incident Central Lines3 lines

{2, 3}, {651, 653}, {662, 811}

▸ Functional Relations3 relations

  • X1981 = bicentric difference of PU(30)

  • X1981 = PU(30)-harmonic conjugate of X1982

  • X1981 = intersection of lines P(15)U(16) and U(15)P(16)

Notes

As a point on the Euler line, X1981 has Shinagawa coefficients \((\$aS_{B}S_{C}\$(E+F)F+\$aS_{A}\$FS^{2} -\$a\$(E-2F)FS^{2}, -\$aS_{B}S_{C}\$S^{2} -3\$aS_{A}\$FS^{2}-\$a\$[(E+F)F-S^{2}]S^{2})\).

If X =\(x : y : z\) is a triangle center other than X1, then the Vega transform of X, defined by trilinears

(y - z)/x : (z - x)/y : (x - y)/z

In general, the bicentrics of X also lie on [\(x:y:z\)]; for details, click Tables at the top of this page.