**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 .

**Comments:** 12 Pages.

**Download:** **PDF**

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

**Unique-IP document downloads:** 600 times

Vixra.org is a pre-print repository rather than a journal. Articles hosted may not yet have been verified by peer-review and should be treated as preliminary. In particular, anything that appears to include financial or legal advice or proposed medical treatments should be treated with due caution. Vixra.org will not be responsible for any consequences of actions that result from any form of use of any documents on this website.

**Add your own feedback and questions here:**

*You are equally welcome to be positive or negative about any paper but please be polite. If you are being critical you must mention at least one specific error, otherwise your comment will be deleted as unhelpful. *