X805

o-(- 2, 2) CIRCUMCIRCLE POINT

X805 = trilinear pole of line X6X694
< 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) { 98, 385 }: Central Line L(98, 385) { 99, 512 }: Central Line L(99, 512) { 110, 669 }: Central Line L(110, 669) { 111, 694 }: Central Line L(111, 694) { 187, 729 }: Central Line L(187, 729) { 249, 827 }: Central Line L(249, 827) A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6122, 0.5999) X_804 = ODD (- 2, 2) INFINITY POINT (0.9171, 0.6474) X_98 = TARRY POINT (0.8891, -0.2281) X_385 = -HIRST INVERSE OF X(6) (2.2863, 2.0795) X_99 = STEINER POINT (0.1109, 0.4322) X_512 = ISOGONAL CONJUGATE OF X(99) (1.0394, -0.1040) X_110 = FOCUS OF KIEPERT PARABOLA (0.2516, 0.5478) X_669 = CROSSDIFFERENCE OF X(2) AND X(39) (-4.3514, 4.6163) X_111 = PARRY POINT (0.9885, -0.0457) X_694 = ISOGONAL CONJUGATE OF X(385) (1.0179, 0.0378) X_187 = INVERSE-IN-CIRCUMCIRCLE OF X(6) (SCHOUTE CENTER) (1.1986, 1.2559) X_729 = EVEN (- 2, 0) CIRCUMCIRCLE POINT (1.0096, 0.1290) X_249 = ISOGONAL CONJUGATE OF X(115) (0.5045, 0.3948) X_827 = o-(0, 4) CIRCUMCIRCLE POINT (0.3242, 0.5811) X_805 = o-(- 2, 2) CIRCUMCIRCLE POINT (0.9729, -0.0897) A B C X804 X98 X385 X99 X512 X110 X669 X111 X694 X187 X729 X249 X827 X805 (a, b, c) = (5, 6, 7) | AB = 1

X805 (o-(- 2, 2) CIRCUMCIRCLE POINT) 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 \(f(a,b,c) = 1/[a ^{-3}(b^{2} - c^{2}) + a^{1}(b ^{-2} - c ^{-2})]\)
Barycentrics: \(g(a,b,c) : g(b,c,a) : g(c,a,b)\), where \(g(a,b,c) = af(a,b,c)\)
Search Value (6, 9, 13): \(0.223824932622\)

X805 lies on the circumcircle.

▸ Incident Central Lines9 lines

{98, 385}, {99, 512}, {110, 669}, {111, 694}, {187, 729}, {249, 827}, {574, 843}, {691, 882}, {888, 892}

▸ Functional Relations10 relations

Notes

Let A'B'C' be the vertex-triangle of the 2nd and 3rd Brocard triangles. The lines AA', BB', CC' concur in X805. (Randy Hutson, March 25, 2016)

Let La, Lb, Lc be the lines through A, B, C, resp. parallel to the Brocard axis. Let Ma, Mb, Mc be the reflections of BC, CA, AB in La, Lb, Lc, resp. Let A' = Mb∩Mc, and define B' and C' cyclically. Triangle A'B'C' is inversely similar to, and 3 times the size of, ABC. The centroid of A'B'C' is X805; see Hyacinthos #16741/16782, Sep 2008. (Randy Hutson, March 25, 2016)

Let La, Lb, Lc be the lines through A, B, C, resp. parallel to the Lemoine axis. Let Ma, Mb, Mc be the reflections of BC, CA, AB in La, Lb, Lc, resp. Let A' = Mb∩Mc, and define B' and C' cyclically. Triangle A'B'C' is inversely similar to, and 3 times the size of, ABC. The centroid of A'B'C' is X805; see Hyacinthos #16741/16782, Sep 2008. (Randy Hutson, March 25, 2016)