Every Triangle is the Shadow of an Equilateral Triangle

Publication Date

Spring 4-16-2025

Presentation Length

20 minutes

College

College of Sciences & Mathematics

Department

Math and Computer Science, Department of

Student Level

Undergraduate

SPARK Category

Scholarship

Faculty Advisor

Adam Cartisano

WELL Core Type

Intellectual Wellness

Metadata/Fulltext

Fulltext

SPARK Session

Geometry Topic

Presentation Type

Talk/Oral

Summary

The Euler line of a triangle is a classical object with an interesting optical illusion: As one translates the vertices of the triangle, the Euler line appears to "pop" into 3D space! The goal of this project is to formalize that intuition by realizing the triangle as the projected image of an equilateral triangle in R^3, where the projection maps the normal line through its centroid to the Euler line of the image.

In this presentation, we will motivate the problem and showcase some of the geometry using GeoGebra activities, and we will prove some partial results: Every triangle in R^2 is the orthogonal projection of an equilateral triangle in R^3, and this projection takes centroids to centroids. We also have a procedure for finding an equilateral triangle which is in perspective with an arbitrary triangle in R^2.

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