## X(110) (FOCUS OF KIEPERT PARABOLA)

 Interactive Applet

You can move the points A, B and C (click on the point and drag it).
Press the keys “+” and “−” to zoom in or zoom out the visualization window and use the arrow keys to translate it.

You can also construct all centers related with this one (as described in ETC) using the “Run Macro Tool”. To do this, click on the icon , select the center name from the list and, then, click on the vertices A, B and C successively.

 The JRE (Java Runtime Environment) is not enabled in your browser!

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

 Information from Kimberling's Encyclopedia of Triangle Centers

Trilinears           csc(B - C) : csc(C - A) : csc(A - B)
= a/(b2 - c2) : b/(c2 - a2) : c/(a2 - b2)

Barycentrics    a2/(b2 - c2) : b2/(c2 - a2) : c2/(a2 - b2)

X(110) - circumcircle-antipode of X(74)
X(110) = isogonal conjugate of the isotomic conjugate of X(99)
X(110) = Ψ(X(6), X(3))
X(110) = Feuerbach point of the tangential triangle.

J. W. Clawson, "Points on the circumcircle," American Mathematical Monthly 32 (1925) 169-174.

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.

Benedetto Scimemi, "Paper-folding and Euler's Theorem Revisited," Forum Geometricorum.

Scimemi proves that if the Euler line is reflected in every side of triangle ABC, then the three reflections concur in X(110).

X(110) lies on these lines:
1,60    2,98    3,74    4,113    5,49    6,111    11,215    20,146    21,104    22,154    23,323    24,155    27,917    28,915    30,477    31,593    32,729    39,755    58,106    65,229    67,141    69,206    81,105    86,675    97,418    99,690    100,643    101,163    102,283    107,648    108,162    143,195    187,352    190,835    249,512    250,520    251,694    274,767    324,436    351,526    353,574    373,575    376,541    476,523    525,935    560,715    595,849    668,839    669,805    670,689    681,823    685,850    789,799    859,953

X(110) is the {X(5),X(49)}-harmonic conjugate of X(54).

X(110) = midpoint of X(I) and X(J) for these (I,J): (3,399), (20,146), (23,323)
X(110) = reflection of X(I) in X(J) for these (I,J): (3,1511), (4,113), (23,1495), (67,141), (74,3), (265,5), (382,1539), (895,6), (1177,206)

X(110) = isogonal conjugate of X(523)
X(110) = isotomic conjugate of X(850)
X(110) = inverse of X(2) in the Brocard circle
X(110) = anticomplement of X(125)
X(110) = X(I)-Ceva conjugate of X(J) for these (I,J): (249,6), (250,3)
X(110) = cevapoint of X(I) and X(J) for these (I,J): (3,520), (5,523), (6,512), (141,525)

X(110) = X(I)-cross conjugate of X(J) for these (I,J):
(1,59), (3,250), (6,249), (109,162), (351,111), (512,6), (520,3), (523,54), (526,74)

X(110) = crosssum of X(I) and X(J) for these (I,J): (2,148), (512,647), (520,647)
X(110) = crossdifference of any two points on line X(115)X(125)
X(110) = X(I)-Hirst inverse of X(J) for these (I,J): (1,245), (2,125), (3,246), (4,247)
X(110) = X(I)-beth conjugate of X(J) for these (I,J): (21,759), (643,643)

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