College Geometry

Some curious properties and loci problems associated with cubics and other polynomials

“…This supports a well-known theorem in geometry, which states that for any triangle, its centroid, circumcentre, orthocentre, and NinePoint centre are always collinear [4,5,10]. A modern day computer algebra system such as Mathematica Õ enables us to discover new results, and at the same time, it can also reinforce old results.…”

“…We would now like to investigate several geometric properties of 4APC, for changing values of t. Some of these geometric properties include the centroid G, circumcentre O, orthocentre H, and centre N of the Nine-Point circle [4,9]. By definition, the circumcentre of a triangle is the point where the perpendicular bisectors of each side intersect.…”

“…For example, let G 1 ð 1 , 1 Þ be the centroid of 4APQ as shown in Figure 2 [4][5][6]. Note that the centroid, or the centre of gravity, is the balancing point of any triangle.…”

An Proposed Algorithm for Generating Criteria Necessary to Establish Congruence between Two Convex N-Sided Polygons in Euclidean Geometry

“…A triangle is the union of three segments (called its sides), whose endpoints (called its vertices) are taken, in pairs, from a set of three noncollinear points. [1] Two triangles are congruent, if they have the same angles and the same sides. Disclaimer/Publisher's Note: The statements, opinions, and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s).…”

“…Two triangles are congruent if under some correspondence between the vertices, the corresponding sides, and corresponding angles are congruent. [2] There are four main criteria to establish congruence between two triangles, they are: [SAS],[SSS],[AAS],[ASA] 1 , where S is a side and A is an angle. Given right angled triangles we can shorten the amount of information necessary, to [LA],…”