## X(355) (FUHRMANN CENTER)

 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.

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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           f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a cos A - (b + c)cos(B - C)

Trilinears           g(a,b,c) : g(b,c,a) : g(c,a,b),
where g(a,b,c) = bc(b + c)[a2(b2 + c2) - (b2 - c2)2] - a3bc(b2 + c2 - a2) (Michel Garitte, 4/3/03)

Barycentrics    af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(355) = 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 shorter arc having endpoints B and C on the circumcircle of ABC; then A" is the reflection of A' in line BC. Vertices B" and C" are obtained cyclically.

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

X(355) lies on these 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

X(355) = midpoint of X(4) and X(8)
X(355) = reflection of X(I) in X(J) for these (I,J): (1,5), (3,10), (944,1385), (1482,946)
X(355) = anticomplement of X(1385)
X(355) = complement of X(944)

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