X1147

ISOGONAL CONJUGATE OF X847

X1147 is the isogonal conjugate of X847.
< 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) { 3, 155 }: Central Line L(3, 155) { 3, 49 }: Central Line L(3, 49) { 2, 54 }: Central Line L(2, 54) { 24, 52 }: Central Line L(24, 52) { 4, 110 }: Central Line L(4, 110) A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6122, 0.5999) X_3 = CIRCUMCENTER (0.5000, 0.1021) X_155 = EIGENCENTER OF ORTHIC TRIANGLE (1.1658, 0.5033) X_49 = CENTER OF SINE-TRIPLE-ANGLE CIRCLE (0.6275, 0.1789) X_2 = CENTROID (0.5374, 0.2000) X_54 = KOSNITA POINT (0.5880, 0.2176) X_24 = PERSPECTOR OF ABC AND ORTHIC-OF-ORTHIC TRIANGLE (0.6627, 0.5278) X_52 = ORTHOCENTER OF ORTHIC TRIANGLE (0.7155, 0.4580) X_4 = ORTHOCENTER (0.6122, 0.3958) X_110 = FOCUS OF KIEPERT PARABOLA (0.2516, 0.5478) X_1147 = ISOGONAL CONJUGATE OF X(847) (0.8329, 0.3027) A B C X3 X155 X49 X2 X54 X24 X52 X4 X110 X1147 (a, b, c) = (5, 6, 7) | AB = 1

X1147 (ISOGONAL CONJUGATE OF X847) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

Trilinears: \((\sin 4A)/\sin A : (\sin 4B)/\sin B : (\sin 4C)/\sin C\)
Barycentrics: \(\sin 4A : \sin 4B : \sin 4C\)
Search Value (6, 9, 13): \(3.259021236949\)
▸ Alternate Coordinates2 forms

  • Trilinears: \(\cos A \cos 2A : \cos B \cos 2B : \cos C \cos 2C\)

  • Barycentrics: \(a^{4} (b^{2} + c^{2} - a^{2}) (a^{4} + b^{4} + c^{4} - 2 a^{2} b^{2} - 2 a^{2} c^{2}) : :\)

▸ Incident Central Lines13 lines

{2, 54}, {3, 49}, {4, 110}, {5, 578}, {24, 52}, {26, 206}, {30, 156}, {55, 1069}, {56, 215}, {140, 141}, {143, 576}, {195, 568}, {912, 960}

▸ Functional Relations16 relations

Notes

V. Thebault, “Sine-triple-angle-circle,” Mathesis 65 (1956) 282-284. (Contributed by Edward Brisse, 3/4/02)

Let A'B'C' be the circumcevian triangle of X4. Let \(R_{A}\) be the radical axis of the circles (B', |B'C|) and (C',|C'B|), and define \(R_{B}\) and \(R_{C}\) cyclically. The lines \(R_{A}\), \(R_{B}\), \(R_{C}\) concur in X1147. For figures, see Concurrent Radical Axes. (Antreas Hatzipolakis and Peter Moses, April 10, 2013)

Let HaHbHc be the orthic triangle. Let A″ be the orthogonal projection of A onto line HbHc, and define B″ and C″ cyclically; then X1147 is the circumcenter of A″B″C″. Let L be the reflection of line HbHc in the perpendicular bisector of segment BC, and define M and N cyclically. Let A* = M∩N, and define B* and C* cyclically; then X1147 is the incenter of A*B*C*. (Randy Hutson, August 26, 2014)

Let OAOBOC be the Kosnita triangle. Let L be the line through OA parallel to the Euler line, and define M and N dyclically. Let L' be the reflection of L in sideline BC, and define M' and N' cyclically. The lines L',M',N' concur in X1147. (Randy Hutson, August 26, 2014)

Let DEF be the anticevian triangle of the circumcircle, O; then X1147 is the centroid of the quadrilateral DEFO. (Randy Hutson, August 26, 2014)

Let A'B'C' be the triangle described in ADGEOM #2697 (8/26/2015, Tran Quang Hung). X1147 = X3-of-A'B'C'. (Randy Hutson, October 15, 2018)

X1147 is the insimilicenter of the circumcircle and the sine-triple-angle circle. (Randy Hutson, December 14, 2014)