Mathematics Project Abstract
PROOFS IN GEOMETRY
Presenter:
Ruozhou Jia, Illinois Mathematics and Science Academy, 1500 West Sullivan Road, Aurora, IL, 60506; joejia@imsa.edu
Advisor:
Dr. Michael Keyton, Illinois Mathematics and Science Academy, 1500 Sullivan Road, Mathematics, Aurora, IL, 60506; 630-907-5967; keyton@imsa.edu
Abstract:
Since the foundation of mathematics, Euclidean geometry has always been an area of great interest. I wish to contribute to geometry by first conjecturing original propositions, then by providing rigorous proofs. The propositions include questions of computation, congruence, collinearity, concurrency, extremum, and invariance. The proofs not only utilize the classical methods of synthetic geometry, but also incorporate techniques of analytical geometry, inversive geometry, complex numbers, and affine transformations. One of the most important results of this Inquiry is that I conjectured and proved that in the nth Euclidean space, where n is an integer greater than 1, the power of a fixed point with respect to a fixed sphere is invariant. Other results include proving that the projections of a vertex of a triangle onto the two interior and two exterior angle bisectors, not emanating from the vertex, are collinear; among all inscribed triangles of a triangle, the orthic triangle has the minimum perimeter; and if an equiangular p-gon has rational sides, where p is a prime number, then the polygon is equilateral.