Sadi Abusaymeh | Yarmouk University of Jordan (original) (raw)

Papers by Sadi Abusaymeh

The Mathematical Gazette, 2018

Also, it is proved in [9] that there exists a point in the plane of for which , , and have the sa... more Also, it is proved in [9] that there exists a point in the plane of for which , , and have the same variance if, and only if, the diagonals are perpendicular to each other. Here the variance of is . Quadrilaterals in which the diagonals are perpendicular to each other are called orthodiagonal, and several of their properties and characterisations can be found in [10]. Again, these characterisations do not include this one. P ABCD PAB PBC PCD PDA ABC a2 + b + c2

Let ABC be a triangle with side-lengths a, b, and c, and with angles A, B, and C. Let AA, BB, and... more Let ABC be a triangle with side-lengths a, b, and c, and with angles A, B, and C. Let AA, BB, and CC be the cevians through a point V , let x, y, and z be the lengths of the segments BA, CB, and AC, and let ξ, η, and ζ be the measures of the angles ∠BAA, ∠CBB, and ∠ACC. The centers V for which x, y, and z are linear forms in a, b, and c are characterized. So are the centers for which ξ, η, and ζ are linear forms in A, B, and C. Let ABC be a non-degenerate triangle with side-lengths a, b, and c, and let V be a point in its plane. Let AA, BB, and CC be the cevians of ABC through V and let the intercepts x, y, and z be defined to be the directed lengths of the segments BA, CB, and AC, where x is positive or negative according as A and C lie on the same side or on opposite sides of B, and similarly for y and z; see Figure 1. To avoid infinite intercepts, we assume that V does not lie on any of the three exceptional lines passing through the vertices of ABC and parallel to the opposite s...

Let the internal angle bisectors BBand CCof angles B and C of triangle ABC be extended to meet th... more Let the internal angle bisectors BBand CCof angles B and C of triangle ABC be extended to meet the circumcircle at B∗ and C∗. The Steiner- Lehmus theorem states that if BB� = CC� , then AB = AC. In this article, we investigate those triangles for which BB∗ = CC∗ and we address several issues that arise within this investigation.

A center function is a function Z that assigns to every triangle T in a Euclidean plane E a point... more A center function is a function Z that assigns to every triangle T in a Euclidean plane E a point Z(T) in E in a manner that is symmetric and that respects isometries and dilations. A family F of center functions is said to be complete if for every scalene triangle ABC and every point P in its plane, there is Z∈ F such that Z(ABC )= P. It is said to be separating if no two center functions in F coincide for any scalene triangle. In this note, we give simple examples of complete separating families of continuous triangle center functions. Regarding the impression that no two different center functions can coincide on a scalene triangle, we show that for every center function Z and every scalene triangle T , there is another center function Z, of a simple type, such that Z(T )= Z(T).

Let ABC be a triangle with side-lengths a, b, and c. For a point P in its plane, let APa, BPb, an... more Let ABC be a triangle with side-lengths a, b, and c. For a point P in its plane, let APa, BPb, and CPc be the cevians through P. It was proved in (1) that the centroid, the Gergonne point, and the Nagel point are the only centers for which (the lengths of) BPa, CPb, and APc are linear forms in

Mathematics Magazine, 1997

Results in Mathematics, 2008

ABSTRACT In this paper, we consider Archimedes’ arbelos and the two identical Archimedean circles... more ABSTRACT In this paper, we consider Archimedes’ arbelos and the two identical Archimedean circles it contains, and we explore possible generalizations to three-dimensional Euclidean space. Thinking of the line segment joining the centers of the two smaller semicircles in the ordinary arbelos as its base, we devise a three-dimensional configuration having a triangle as base and containing three spheres that seem to play the roles of the two Archimedean circles. In Theorem 3.1, we find formulas for the radii of these spheres (in terms of the base triangle) and conditions under which two of them are equal, and we describe (in the last paragraph of Section 1) how these conditions can be viewed as an honest generalization of the ordinary arbelos theorem. These conditions also imply that the three spheres are equal if and only if the base triangle is equilateral. As a by-product, Theorem 3.4 shows that there is associated with the ordinary arbelos a pair of spheres that amazingly turn out to have the same radii as the Archimedean circles.

Journal of Geometry, 2012

ABSTRACT Propositions 24 and 25 of Book I of Euclid’s Elements state the fairly obvious fact that... more ABSTRACT Propositions 24 and 25 of Book I of Euclid’s Elements state the fairly obvious fact that if an angle in a triangle is increased (without changing the lengths of its arms), then the length of the opposite side increases. In less technical terms, the wider you open your mouth, the farther apart your lips are. In this paper, we see that this has a very satisfactory analogue for orthocentric (but not for general) tetrahedra.

International Journal of Mathematical Education in Science and Technology, 2005

... previous section with a new understanding of the variables involved. As before, we start with... more ... previous section with a new understanding of the variables involved. As before, we start witha general triangle ABC and we take points A′, B′ and C′ on its sides. This time, however, we let x, X, y, Y, z, and Z be defined as shown in figure 2 by. ... see [9, Theorem 1.15.3, p. 56]. ...

Elemente der Mathematik, 2012

Communications in Algebra, 1995