Constructing a quadrilateral inside another one

TL;DR

This paper investigates the geometric properties and area ratios of an interior quadrilateral formed by connecting vertices of a convex quadrilateral to midpoints or points dividing sides in a fixed ratio, exploring how these ratios change with different division points.

Contribution

It introduces a generalized method for constructing an interior quadrilateral based on side division ratios and analyzes the resulting area ratios, extending classical midpoint constructions.

Findings

The area ratio of the interior quadrilateral to the original quadrilateral is characterized.

The effect of varying the division ratio rho on the area ratio is determined.

Explicit formulas for area ratios under different side division ratios are provided.

Abstract

Connect each vertex of a convex quadrilateral Q to the midpoint of the next (proceeding counterclockwise) side. The four connecting lines create an interior quadrilateral I. We study the ratio area(I)/area(Q). We also determine what happens to area(I)/area(Q) when the four midpoints are replaced by points which divide the sides in the ratio of rho to (1-rho) proceeding clockwise. Here rho is any fixed number satisfying 0 < rho < 1.