INTERNATIONAL JOURNAL OF ARTIFICIAL INTELLIGENCE
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THE APPLICATION OF PROJECTIVE GEOMETRY IN ELEMENTARY
GEOMETRY PROBLEMS
Ravshanova O'g'ilshod Abdurashid kizi
Termiz State Pedagogical Institute
Faculty of Natural and Exact Sciences
Student of Mathematics and Informatics Department
Abstract:
This article examines the role 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
the understanding and problem-solving skills of students 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 to solve geometric problems and reveals deeper connections 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
establish key 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 their use through typical problem examples. The goal is to highlight
how projective geometry complements and extends Euclidean methods, offering new insights
and efficient solutions in the study and teaching of geometry.
INTERNATIONAL JOURNAL OF ARTIFICIAL INTELLIGENCE
ISSN: 2692-5206, Impact Factor: 12,23
American Academic publishers, volume 05, issue 08,2025
Journal:
https://www.academicpublishers.org/journals/index.php/ijai
197
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 into more manageable or even 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 configurations and allows for more elegant 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 without changing the underlying incidence properties.
Another powerful tool is the cross-ratio, defined for four collinear points A,B,C,A, B, C,A,B,C,
and DDD, 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
configurations by relating them to simpler or well-understood cases.
The cross-ratio plays a significant role in problems involving division of segments and the
concurrency of lines. For example, in harmonic division—a special case of cross-ratio equal to -
1—the four points A,B,C,DA, B, C, DA,B,C,D satisfy (A,B;C,D)=−1(A, B; C, D) = -
1(A,B;C,D)=−1. Harmonic conjugates have important geometric properties and appear in
theorems related to angle bisectors, cevians, and circle power.
Projective geometry also elegantly handles the concept of poles and polars with respect to a
conic section, typically a circle in elementary geometry. The pole-polar relationship establishes
a duality between points and lines, allowing transformations of problems involving tangents,
chords, and intersections into their dual statements. This duality is particularly useful when
dealing with circle geometry problems that are cumbersome to approach through metric
methods.
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ISSN: 2692-5206, Impact Factor: 12,23
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For example, the famous Pascal’s theorem, which states that the intersection points of the pairs
of opposite sides of a hexagon inscribed in a conic lie on a straight line, is fundamentally a
projective result. When applied to circles, it simplifies to properties about chords and secants
that can be used to solve seemingly complicated problems in elementary geometry.
By leveraging projective methods, problems involving parallelism, concurrency, and
collinearity become more tractable. For example, the Desargues theorem—a central result in
projective geometry—asserts that if two triangles are perspective from a point, then they are
perspective from a line, and vice versa. While it may appear abstract, this theorem has concrete
applications in proving the concurrency of lines in planar geometry problems.
In practical problem solving, projective transformations can be used to map complicated
geometric configurations into simpler ones. Since projective properties are invariant, proving a
statement for a simplified configuration suffices to establish it in the general case. For example,
one can map a given triangle to an equilateral triangle or arrange lines so that certain points
align conveniently, perform calculations or proofs, and then apply the inverse transformation to
conclude the original problem.
This approach contrasts with Euclidean methods that often require direct and sometimes
lengthy algebraic or trigonometric computations. Projective geometry’s flexibility reduces
complexity and increases elegance in proofs and constructions.
Moreover, projective geometry bridges the gap between synthetic and analytic geometry. While
synthetic geometry emphasizes direct constructions and visual reasoning, projective methods
can be combined with coordinate systems—such as homogeneous coordinates—to facilitate
algebraic manipulation of geometric objects. Homogeneous coordinates represent points in the
projective plane using triples (x,y,w)(x, y, w)(x,y,w), where the usual Cartesian coordinates
correspond to (x/w,y/w)(x/w, y/w)(x/w,y/w) when w≠0w \neq 0w =0, and points at infinity
are represented when w=0w=0w=0. This framework allows the use of linear algebra techniques
in solving geometric problems.
In the educational context, introducing projective geometry concepts enriches students’
understanding of geometry by providing new perspectives and tools. Problems that may seem
difficult or cumbersome in classical Euclidean settings often become accessible and instructive
when approached projectively. This enhances both problem-solving skills and appreciation for
the unity and beauty of geometry.
An illustrative example involves the use of projective geometry in solving problems about the
concurrency of cevians in a triangle. For instance, proving Ceva’s theorem through projective
methods highlights the role of incidence and ratio invariance, avoiding complex angle
computations.
Additionally, projective methods help in understanding and proving properties of special points
and lines associated with triangles, such as the centroid, orthocenter, and circumcenter, when
considered in a projective setting. The generalization to conics instead of just circles opens
avenues for exploring deeper geometric relationships.
INTERNATIONAL JOURNAL OF ARTIFICIAL INTELLIGENCE
ISSN: 2692-5206, Impact Factor: 12,23
American Academic publishers, volume 05, issue 08,2025
Journal:
https://www.academicpublishers.org/journals/index.php/ijai
199
In summary, projective geometry’s emphasis on incidence and invariance under projection
transforms classical geometry problems into more general and sometimes simpler forms. By
introducing points at infinity, employing cross-ratio and harmonic division, and utilizing
projective transformations, many elementary geometry problems gain elegant and unified
solutions. This not only broadens mathematical insight but also fosters creative and flexible
problem-solving approaches in both education and research.
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Geometry: Euclid and Beyond
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Geometry Revealed: A Jacob’s Ladder to Modern Higher Geometry
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A Treatise on the Higher Plane Curves
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