**Authors:** Markos Georgallides

In this work is given a new approach to the Open Question of professor Florentine Smarandache concerning the decreasing Tunnel for Orthocenter H on any triangle ABC . Circumcenter O , Centroid K and Ortocenter H lie on Euler line OH . The midpoint N of segment OH is the center of the nine - points circle which is passing from the three midpoints of each side and from the three feet of the altitudes , so this point N is orthic`s triangle circum center . This property of point N ( as it is the first link of a chain ) connects segment ( bar ) OH with an infinite set of segments OnHn of the orthic triangles where On coincides with point Nn-1 , that of each time midpoint of segments . This chain is the locus of point N and that of the repetitive ( rotating ) segment OnHn . On any triangle ABC and on the vertices of the triangle , is constructed an orthogonal hyperbola which passes from orthocenter and provides two fix points ( the foci ) in plane . As a result is the Axial Symmetry to the two axis , the orthogonal x,y and that of asymptotes . Since orthocenter H changes position , then AH is altering magnitude and direction , therefore AH is a repetitive damped Vector Quantity which assumes its extreme in the opposite direction relative to the first or prior positions . The above property results to a Central Symmetry to one of the vertices A , B , C with the two hyperbolas and after following the greatest of sides a , b , c . Damped Vector AHn can then convergent to Hn which is the Orthocenter of AnBnCn and it is the extreme in opposite direction . i.e. Orthocenter H… Hn limits to a point on a chain ( straight line or curved ) through A .

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[v1] 2012-01-10 09:29:35

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