[3] **viXra:1201.0090 [pdf]**
*replaced on 2013-02-10 20:57:47*

**Authors:** Morio Kikuchi

**Comments:** 10 Pages.

In the inversion between two coordinate spheres, a ratio of length and constant of length is invariable.

**Category:** Geometry

[2] **viXra:1201.0061 [pdf]**
*submitted on 2012-01-15 22:14:03*

**Authors:** Florentin Smarandache, Ion Patrascu

**Comments:** 244 Pages.

This book is addressed to students, professors and researchers of
geometry, who will find herein many interesting and original results.
The originality of the book The Geometry of Homological Triangles
consists in using the homology of triangles as a “filter” through which
remarkable notions and theorems from the geometry of the triangle are
unitarily passed.
Our research is structured in seven chapters, the first four are
dedicated to the homology of the triangles while the last ones to their
applications.

**Category:** Geometry

[1] **viXra:1201.0047 [pdf]**
*submitted on 2012-01-10 09:29:35*

**Authors:** Markos Georgallides

**Comments:** 12 Pages.

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 .

**Category:** Geometry