## X(125) (CENTER OF JERABEK HYPERBOLA)

 Interactive Applet

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This applet was built with the free and multiplatform dynamic geometry software C.a.R..

 Information from Kimberling's Encyclopedia of Triangle Centers

Trilinears           cos A sin2(B - C) : cos B sin2(C - A) : cos C sin2(A - B)
= (sec A)(c cos C - b cos B)2 : (sec B)(a cos A - c cos C)2 : (sec C)(b cos B - a cos A)2
= bc(b2 + c2 - a2)(b2 - c2)2 : ca(c2 + a2 - b2)(c2 - a2)2 : ab(a2 + b2 - c2)(a2 - b2)2

Barycentrics    (sin 2A)[sin(B - C)]2 : (sin 2B)[sin(C - A)]2 : (sin 2C)[sin(A - B)]2

X(125) lies on the nine-point circle
X(125) = X(110)-of-medial triangle
X(125) = X(100)-of-orthic triangle, if ABC is acute

Roland H. Eddy and R. Fritsch, "The conics of Ludwig Kiepert: a comprehensive lesson in the geometry of the triangle," Mathematics Magazine 67 (1994) 188-205.

X(125) lies on these lines:
2,98    3,131    4,74    5,113    6,67    51,132    68,1092    69,895    115,245    119,442    122,684    126,141    128,140    136,338    381,541    511,858

X(125) = midpoint of X(I) and X(J) for these (I,J): (3,265), (4,74), (6,67)
X(125) = reflection of X(I) in X(J) for these (I,J): (113,5), (185,974), (1495,468), (1511,140), (1539,546)
X(125) = isogonal conjugate of X(250)
X(125) = inverse-in-Brocard-circle of X(184)
X(125) = complement of X(110)
X(125) = complementary conjugate of X(523)
X(125) = X(I)-Ceva conjugate of X(J) for these (I,J): (4,523), (66,512), (68,520), (69,525), (338,115)
X(125) = crosspoint of X(I) and X(J) for these (I,J): (4,523), (69,525), (338,339)
X(125) = crosssum of X(I) and X(J) for these (I,J): (3,110), (25,112), (162,270), (1113,1114)
X(125) = crossdifference of any two points on line X(110)X(112)
X(125) = X(115)-Hirst inverse of X(868)
X(125) = X(2)-line conjugate of X(110)

This is a joint work of
Humberto José Bortolossi, Lis Ingrid Roque Lopes Custódio and Suely Machado Meireles Dias.