Max/Min Puzzles in Geometry

TL;DR

This paper investigates the problem of determining the largest inscribed polygons, such as parallelograms, rectangles, and squares, within a triangle, and extends the analysis to include wedged polygons.

Contribution

It introduces a method to find maximum-area polygons inscribed in a triangle and generalizes the concept to wedged polygons, advancing geometric optimization techniques.

Findings

Identifies maximum-area inscribed parallelograms, rectangles, and squares within a triangle.

Extends the analysis to include wedged polygons.

Provides geometric bounds and optimization strategies.

Abstract

The objective here is to find the maximum polygon, in area, which can be enclosed in a given triangle, for the polygons: parallelograms, rectangles and squares. It will initially be assumed that the choices are inscribed polygons, that is all vertices of the polygon are on the sides of the triangle. This concept will be generalized later to include wedged polygons.