fairfolio
Saved Projects

Projective Generalization of Foci and Isogonal Cubics Loci of Complete Quadrilaterals

ISEF · 2026 Mathematics

Lu, Pin-Chun · Taipei Municipal Chien Kuo High School

Overview

The theory of isogonal cubics is a powerful tool in proving the relations of angles. A part of the theory is deduced from Newton's theorems on conics. It is known that Newton's theorems can be generalized, so it's a natural question to ask if the entire theory of isogonal cubics can be generalized. In this study, we used projective geometry to describe and deduce properties of conics, which gave a whole new perspective to the Euclidean definition of conics and related objects. We start from generalizing fundamental geometrical objects such as the foci of conics, pedal circles and Miquel points projectively. Motivated by these generalizations, we proved that for a point, there exist an isoconjugate of it regarding the complete quadraliteral if and only if it satisfies a certain cross ratio equation. Finally, we proved that an isoconjugation is equivalent to an isoconjugation on a specific triangle, we used this result to prove the locus of isoconjugation is a cubic.

Awards (1)

  • Fourth Award of $600 $600

Competition history

  • ISEF 2026 Mathematics · Entry MATH008 Affiliated fair in Taiwan

Resources

Related projects

Closest projects by meaning, across every fair and year in the corpus.

Browse more like this

Source: Regeneron International Science and Engineering Fair

Save projects to your library

Sign in with Google to keep track of projects you find interesting, organized into folders. An account also raises your daily allowance for “Has this been done?”, and lets you create a key for the MCP server with a much higher limit than anonymous use. Browsing stays public.

Continue with Google