X944

HOFSTADTER TRAPEZOID POINT

X944 is the point in which the extended legs X4X1 and X3X8 of the trapezoid X4X1X3X8 meet. The point is introduced in
< 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) { 20, 145 }: Central Line L(20, 145) { 1, 4 }: Incenter–Orthocenter Line { 3, 8 }: Central Line L(3, 8) { 48, 281 }: Central Line L(48, 281) { 2, 355 }: Central Line L(2, 355) A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6122, 0.5999) X_20 = DE LONGCHAMPS POINT (0.3878, -0.1916) X_145 = ANTICOMPLEMENT OF NAGEL POINT (0.6735, 0.3333) X_1 = INCENTER (0.5714, 0.2333) X_4 = ORTHOCENTER (0.6122, 0.3958) X_3 = CIRCUMCENTER (0.5000, 0.1021) X_8 = NAGEL POINT (0.4694, 0.1333) X_48 = CROSSPOINT OF X(1) AND X(63) (0.5412, 0.1246) X_281 = -CROSS CONJUGATE OF X(9) (0.5796, 0.3202) X_2 = CENTROID (0.5374, 0.2000) X_355 = FUHRMANN CENTER (0.5408, 0.2645) X_944 = HOFSTADTER TRAPEZOID POINT (0.5306, 0.0708) A B C X20 X145 X1 X4 X3 X8 X48 X281 X2 X355 X944 (a, b, c) = (5, 6, 7) | AB = 1

X944 (HOFSTADTER TRAPEZOID POINT) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

Trilinears: \(bc[3a^{4} - 2a^{3}(b + c) + (b - c)^{2}(2ab + 2ac - 2bc - b^{2} - c^{2} - 2a^{2})]\)
Search Value (6, 9, 13): \(9.057236429841\)
▸ Alternate Coordinates2 forms

  • Trilinears: \(1 - \cos A - \cos B - \cos C + \cos B \cos C\)

  • Trilinears: \(r - R \cos B \cos C : :\)

▸ Linear Combinations1 combination
▸ Incident Central Lines16 lines

{1, 4}, {2, 355}, {3, 8}, {5, 3616}, {10, 631}, {20, 145}, {30, 962}, {40, 376}, {48, 281}, {80, 499}, {84, 1000}, {150, 348}, {390, 971}, {392, 452}, {938, 999}, {958, 1006}

▸ Functional Relations10 relations

Notes

Hofstadter, Douglas. R., “Discovery and dissection of a geometric gem,” in Geometry Turned On! editors J. R. King and D. Schattschneider, Mathematical Association of America, Washington, D. C., 1997, 3-14.

The centroid of ABC is also the centroid of triangle X4X8X944. (Darij Grinberg, August 22, 2002)

Let Ha be the hyperbola passing through A, and with foci at B and C. This is called the A-Soddy hyperbola in (Paul Yiu, Introduction to the Geometry of the Triangle, 2002; 12.4 The Soddy hyperbolas, p. 143). Let La be the polar of X3 with respect to Ha. Define Lb and Lc cyclically. Let A' = Lb ∩ Lc, B' = Lc ∩ La, C' = La ∩ Lb. Triangle A'B'C' is homothetic to ABC, and its orthocenter is X944. (Randy Hutson, January 29, 2018)

Let Ma be the polar of X4 wrt the circle centered at A and passing through X1, and define Mb, Mc cyclically. (Note: X4 is the perspector of any circle centered at a vertex of ABC.) Let A″ = Mb ∩ Mc, and define B″ and C″ cyclically. Triangle A″B″C″ is homothetic to ABC, and its orthocenter is X944. (Randy Hutson, January 29, 2018)