Some properties of rectangle and a random point

TL;DR

This paper explores geometric properties involving rectangles and random points, establishing relationships between key triangle centers and demonstrating the effectiveness of Cartesian coordinates in solving related problems.

Contribution

It introduces new theorems linking the Euler line and Brocard axis in rectangle configurations and applies Cartesian coordinates to solve related geometric problems.

Findings

Established a relationship between Euler line and Brocard axis in rectangle configurations

Proposed and solved related geometric problems using Cartesian coordinates

Demonstrated the utility of Cartesian coordinates in geometric theorem proving

Abstract

We establish a relationship between the two important central lines of the triangle, the Euler line and the Brocard axis, in a configuration with an arbitrary rectangle and a random point. The classical Cartesian coordinate system method shows its strength in these theorems. Along with that, some related problems on rectangles and a random point are proposed with similar solutions using Cartesian coordinate system.