Topic
Nine-point center
About: Nine-point center is a research topic. Over the lifetime, 3 publications have been published within this topic receiving 3 citations.
Papers
1 Mar 2012
TL;DR: In this article, the authors consider the problem of partitioning the plane into two sets containing their triangle centers and show that S must be dense in the plane, and also consider several problems about partitioning S.
Abstract: Let S be a set of at least five points in the plane, not all on a line. Suppose that for any three points \({a,b,c\in S}\) the nine-point center of triangle abc also belongs to S. We show that S must be dense in the plane. We also consider several problems about partitioning the plane into two sets containing their triangle centers.
3 citations
1 Jan 2012
TL;DR: In this article, the problem of constructing a triangle given its circumcenter, incenter, and one vertex is revisited, and it is established that such a triangle exists if and only if the incenter lies inside the cardioid relative to the circumcircle.
Abstract: We construct a triangle given its incenter, nine-point center and a vertex by locating the circumcenter as an intersection of two rectangular hyperbo- las. Some special configurations leading to solutions constructible with ruler and compass are studied. The related problem of construction of a triangle given its circumcenter, incenter, and one vertex is revisited, and it is established that such a triangle exists if and only if the incenter lies inside the cardioid relative to the circumcircle.
1 citations
1 Jan 2014
TL;DR: In this paper, the authors give a simple proof of Gibert's generalization of the Lester circle theorem, which states that every circle whose diameter is a chord of the Kiepert hyperbola perpendicular to the Euler line passes through the Fermat points.
Abstract: We give a simple proof of Gibert's generalization of the Lester circle theorem. The famous Lester circle theorem states that for a triangle, the two Fermat points, the nine point center and the circumcenter lie on a circle, the Lester circle of the triangle. Here is Gibert's generalization of the Lester circle theorem, given in (2) and (4, Theorem 6): Every circle whose diameter is a chord of the Kiepert hyperbola perpendicular to the Euler line passes through the Fermat points. In this note we show that this follows from a property of rectangular hyperbolas. Lemma 1. Let F+ and Fbe two antipodal points on a rectangular hyperbola. For every point H on the hyperbola, the tangent to the circle (F+FH) at H is parallel to the tangents of the hyperbola at F+ and F�.
Performance Metrics
| Year | Papers |
|---|---|
| 2014 | 1 |
| 2012 | 2 |