X355

FUHRMANN CENTER

X355 = the center of the Fuhrmann circle, defined as the circumcircle of the Fuhrmann triangle A″B″C″, where A″ is obtained as follows: let A' be the midpoint of the circumcircle-arc having endpoints B and C and not containing A; then A″ is the reflection of A' in line BC. Vertices B″ and C″ are obtained cyclically. (Other constructions of A″, and hence the Fuhrmann triangle, follow: (1) Let IA be the reflection of X1 in BC; then A″ is the circumcenter of IABC. (2) Let JA be the reflection of the A-excenter in BC; then A″ is the circumcenter of JABC.
< 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) { 4, 8 }: Fuhrmann Line { 1, 5 }: Feuerbach Line { 3, 10 }: Central Line L(3, 10) { 30, 40 }: Central Line L(30, 40) { 65, 68 }: Central Line L(65, 68) A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6122, 0.5999) X_4 = ORTHOCENTER (0.6122, 0.3958) X_8 = NAGEL POINT (0.4694, 0.1333) X_1 = INCENTER (0.5714, 0.2333) X_5 = NINE-POINT CENTER (0.5561, 0.2489) X_3 = CIRCUMCENTER (0.5000, 0.1021) X_10 = SPIEKER CENTER (0.5204, 0.1833) X_30 = EULER INFINITY POINT (0.7469, 0.7481) X_40 = BEVAN POINT (0.4286, -0.0292) X_65 = ORTHOCENTER OF THE INTOUCH TRIANGLE (0.6041, 0.2933) X_68 = PRASOLOV POINT (-0.0535, -0.0055) X_355 = FUHRMANN CENTER (0.5408, 0.2645) A B C X4 X8 X1 X5 X3 X10 X30 X40 X65 X68 X355 (a, b, c) = (5, 6, 7) | AB = 1

X355 (FUHRMANN CENTER) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

Trilinears: \(a \cos A - (b + c)\cos(B - C) : :\)
Search Value (6, 9, 13): \(-0.584565026187\)
▸ Alternate Coordinates3 forms

  • Trilinears: \(bc(b + c)[a^{2}(b^{2} + c^{2}) - (b^{2} - c^{2})^{2}] - a^{3}bc(b^{2} + c^{2} - a^{2}) : :\) (Michel Garitte, 4/3/03)

  • Trilinears: \(\cos B + \cos C - 2 \cos B \cos C - 1 : :\)

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

▸ Incident Central Lines13 lines

{1, 5}, {2, 944}, {3, 10}, {4, 8}, {30, 40}, {65, 68}, {85, 150}, {104, 404}, {165, 550}, {381, 519}, {382, 516}, {388, 942}, {938, 1056}

▸ Functional Relations19 relations

  • X355 = the center of the Fuhrmann circle, defined as the circumcircle of the Fuhrmann triangle A″B″C″, where A″ is obtained as follows: let A' be the midpoint of the circumcircle-arc having endpoints B and C and not containing A; then A″ is the reflection of A' in line BC. Vertices B″ and C″ are obtained cyclically. (Other constructions of A″, and hence the Fuhrmann triangle, follow: (1) Let IA be the reflection of X1 in BC; then A″ is the circumcenter of IABC. (2) Let JA be the reflection of the A-excenter in BC; then A″ is the circumcenter of JABC.

  • X355 = midpoint of X4 and X8

  • X355 = isogonal conjugate of X3417

  • X355 = reflection of Xi in Xj for these (i,j): (1, 5), (3, 10), (944, 1385), (1482, 946)

  • X355 = anticomplement of X1385

  • X355 = complement of X944

  • X355 = inverse-in-Feuerbach-hyperbola of X1837

  • X355 = {X1,X80}-harmonic conjugate of X1837

  • X355 = outer-Garcia-isogonal conjugate of X40

  • X355 = X3-of-outer-Garcia-triangle

  • X355 = X6265-of-inner-Garcia-triangle

  • X355 = X6102-of-excentral-triangle

  • X355 = inner-Garcia-to-outer-Garcia similarity image of X3

  • X355 = X12844-of-orthic-triangle if ABC is acute

  • X355 = endo-homothetic center of Ehrmann side-triangle and 4th anti-Euler triangle; the homothetic center is X12111

  • X355 = reflection of X12699 in X4

  • X355 = {X1,X1837}-harmonic conjugate of X5722

  • X355 = Ursa-minor-to-Ursa-major similarity image of X1

  • X355 = {X1,X5}-harmonic conjugate of X5886

Notes

X355 is the homothetic center of the triangles formed by internal and external tangents of circumcircles of BCX4, CAX4, ABX4. Equivalently, the incenter of the trianglar hull of circumcircles of BCX4, CAX4, ABX4. (Randy Hutson, December 2, 2017)

Let Na = X5-of-BCX1, Nb = X5-of-CAX1, Nc = X5- of-ABX1. Then X355 = X20-of-NaNbNc. (Randy Hutson, December 2, 2017)

Let KaKbKc be the anticomplementary triangle. Then X355 is the radical center of the incircles of KaBC, KbCA, KcAB. (Randy Hutson, December 2, 2017)

Let A'B'C' be the outer Garcia triangle and A″B″C″ the inner Garcia triangle. Let A* be the isogonal conjugate, wrt A'B'C', of A″, and define B* and C* cyclically. The lines A'A*, B'B*, C'C* concur in X355; see also X952. (Randy Hutson, December 2, 2017)

Ross Honsberger, Episodes in Nineteenth and Twentieth Century Euclidean Geometry, Mathematical Association of America, 1995. Chapter 6: The Fuhrmann Circle.

Let OA be the circle centered at the A-vertex of the Yff contact triangle and passing through A; define OB and OC cyclically. X355 is the radical center of OA, OB, OC. (Randy Hutson, August 30, 2020)