THE APPLICATION OF PROJECTIVE GEOMETRY IN ELEMENTARY GEOMETRY PROBLEMS

Аннотация

This article examines therole of projective geometry in solving elementary geometry problems. Projective geometry, a branch of mathematics focusing on properties invariant under projection, provides powerful tools and perspectives that simplify and generalize classical geometric constructions and proofs. By extending the Euclidean plane to include points at infinity and employing concepts such as cross-ratio and harmonic division, projective methods enable elegant solutions to problems involving collinearity, concurrency, and incidence relations. The paper illustrates key projective geometry principles and demonstrates their applications through typical elementary geometry problems, highlighting how projective approaches can unify and enrich traditional Euclidean techniques. This study aims to enhance theunderstanding and problem-solvingskills ofstudents and educators in geometry. Keywords: Projective geometry, elementary geometry, collinearity, concurrency, incidence relations, cross-ratio, harmonic division, points at infinity, geometric transformations, Euclidean geometry. Elementary geometry, rooted in Euclid’s axioms, studies the properties and relations of points, lines, and figures on the Euclidean plane. While classical methods rely heavily on metric concepts such as distances and angles, projective geometry offers an alternative framework that focuses on properties invariant under projection. This shift in perspective broadens the toolkit available tosolvegeometric problems and reveals deeperconnections among geometric figures. Projective geometry introduces concepts such as points at infinity, which unify parallel lines by considering their intersection “at infinity.” This extension simplifies many proofs and constructions that are cumbersome in purely Euclidean terms. Fundamental notions like the cross-ratio of four collinear points and harmonic division provide powerful invariants that help establishkey incidence and concurrency properties. The application of projective geometry in elementary problems is not only a theoretical enrichment but also a practical method to tackle classical questions involving lines, circles, and polygons. By integrating projective concepts, problem solvers can approach geometry with greater flexibility and elegance, often reducing complex configurations to simpler projective relations. This paper explores the essential elements of projective geometry relevant to elementary geometry and demonstrates theiruse through typical problem examples. The goal is to highlight how projective geometry complements and extends Euclidean methods, offering new insights andefficient solutionsin thestudy andteaching ofgeometry. 196 INTERNATIONAL JOURNAL OF ARTIFICIAL INTELLIGENCE ISSN:2692-5206,ImpactFactor:12,23 AmericanAcademicpublishers,volume05,issue08,2025 Journal: https://www.academicpublishers.org/journals/index.php/ijai Projective geometry, unlike classical Euclidean geometry, studies properties of figures that remain invariant under projection transformations. Its focus on incidence and alignment rather than measurements such as lengths and angles enables mathematicians and students to approach elementary geometry problems with novel and powerful tools. The introduction of concepts such as points at infinity, cross-ratio, and harmonic division transforms many classical problems intomore manageable oreven straightforward ones. One of the foundational ideas in projective geometry is the extension of the Euclidean plane to the projective plane by adding “points at infinity” corresponding to directions of parallel lines. In Euclidean geometry, parallel lines never intersect, which often complicates proofs involving concurrency or collinearity. In the projective plane, however, parallel lines meet at a unique point at infinity, unifying the treatment of parallel and intersecting lines. This unification simplifies many geometric configurationsand allows for moreelegant proofs. For instance, consider the problem of proving the concurrency of three cevians in a triangle, such as medians or altitudes. In Euclidean geometry, such proofs typically involve intricate angle chasing or length ratio arguments. In projective geometry, concurrency is treated through incidence relations that remain valid under projection, often simplifying the argument. The concept of the projective transformation allows the repositioning of points and lines to more convenient configurations withoutchanging theunderlying incidence properties. Another powerful tool is the cross-ratio, defined for four collinear points A,B,C,A, B, C,A,B,C, andDDD, as (A,B;C,D)=AC⋅ BDAD⋅ BC,(A, B; C, D) = \frac{AC \cdot BD}{AD \cdot BC},(A,B;C,D)=AD⋅ BCAC⋅ BD​ , where the segments represent signed distances. The cross-ratio is invariant under projective transformations, meaning it remains constant regardless of the perspective from which the points are viewed. This invariance enables problem solvers to analyze complex point configu