Tangent Spheres and Triangle Centers

A simplex is said to be orthocentric if its altitudes intersect in a common point, called its orthocenter. In this paper it is proved that if any two of the traditional centers of an orthocentric simplex (in any dimension) coincide, then the simplex is regular. Along the way orthocentric simplices in which all facets have the same circumradius are characterized, and the possible barycentric coordinates of the orthocenter are described precisely. In particular these barycentri...

It is known that for each combinatorial type of convex 3-dimensional polyhedra, there is a representative with edges tangent to the unit sphere. This representative is unique up to projective transformations that fix the unit sphere. We show that there is a unique representative (up to congruence) with edges tangent to the unit sphere such that the origin is the barycenter of the points where the edges touch the sphere.

In this paper, we present a novel method to draw a circle tangent to three given circles lying on a plane. Using the analytic geometry and inversion (reflection) theorems, the center and radius of the inversion circle are obtained. Inside any one of the three given circles, a circle of the similar radius and concentric with its own corresponding original circle is drawn.The tangent circle to these three similar circles is obtained. Then the inverted circles of the three simil...

In this paper, we present a synthetic solution to a geometric open problem involving the radical axis of two strangely defined circumcircles. The solution encapsulates two generalizations, one of which uses a powerful projective result relating isogonal conjugation and polarity with respect to circumconics.

We present a class of explicit solutions for the problem of minimization of the function $f(x,y,z)=\sum_{i=1}^{4}\sqrt{(x-x_{i})^2+(y-y_{i})^2+(z-z_{i})^2},$ which gives the location of the unique stationary (Fermat-Torricelli) point for four non-collinear and non-coplanar points $A_{i}=(x_{i},y_{i},z_{i}),$ determining tetrahedra, which are derived by a proper class of isosceles tetrahedra having four equal edges and two equal opposite edges. This class of explicit solutions...

We explicitly construct small triangulations for a number of well-known 3-dimensional manifolds and give a brief outline of some aspects of the underlying theory of 3-manifolds and its historical development.

We present generalizations of theorems on Kypert's construction and on 2nd Morley's Centre. Most of our proofs are synthetic.

We solve the following geometric problem, which arises in several three-dimensional applications in computational geometry: For which arrangements of two lines and two spheres in R^3 are there infinitely many lines simultaneously transversal to the two lines and tangent to the two spheres? We also treat a generalization of this problem to projective quadrics: Replacing the spheres in R^3 by quadrics in projective space P^3, and fixing the lines and one general quadric, we g...

The setting for this brief paper is R^3. Distance between two spheres is understood as distance delta between spherical centers. For instance, a Reuleaux tetrahedron T is the intersection of four unit balls satisfying delta=1 pairwise. Volume and surface area of T are already well-known; our humble contribution is to calculate the mean width of T.

In this article we give new proofs for the existence and basic properties of the cirucmcenter of mass defined by V. E. Adler in 1993 and S. Tabachnikov and E. Tsukerman in 2013.