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Exploring Euclidean Geometry with Python: A Practical Guide

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Chapter 1: Introduction to Scientific Python

In this series, we delve into the practical applications of the Python scientific stack for tackling real-world challenges. Mastering these tools takes consistent practice, and this series aims to guide you through solving both common and unusual problems using libraries such as NumPy, SymPy, SciPy, and matplotlib.

Today's Challenge

Today, we will focus on creating geometric shapes using matplotlib while also leveraging vector algebra with SymPy. The objective is to demonstrate Thales' theorem without traditional pen and paper, relying solely on Python's scientific libraries. Although the theorem itself is straightforward, the emphasis here is on efficiently utilizing the Python stack.

Geometric representation of Thales' theorem

Thales' Theorem Explained

Thales' theorem asserts that if a triangle's one side is the diameter of a circle, and the third point lies on the circumference, then the angle opposite to the diameter is a right angle. To visualize this scenario, we will plot it using a Jupyter notebook. Below is a plotting script with comments for clarity.

To prove Thales' theorem using vector algebra, we will establish a coordinate system centered at the circle's origin. We will also add coordinate axes to our plot for better understanding.

Circle with triangle demonstrating Thales' theorem

With our coordinate system in place, we can now plot the position vectors for points A, B, and C.

Position vectors for points A, B, and C

Using SymPy for Proof

For the proof, we will define the polar angles for points A and B as 0 and θ, respectively, while allowing the angle at point C to be arbitrary. The radius of the circle will also be treated as a variable. We will define the necessary symbols and vectors as follows:

Symbols and vectors for the proof

Our goal is to establish that the angle at point C is indeed a right angle. To do this, we will need to calculate the sides AC and BC.

Sides AC and BC

Next, we compute the scalar product:

Scalar product calculation

Since the scalar product is zero, we can conclude that the angle at point C is a right angle.

Chapter 2: Practical Applications and Further Exploration

This video titled "Euclidean Distance - Practical Machine Learning Tutorial with Python" explores how to compute distances in Euclidean space using Python, providing practical examples and coding insights.

In this follow-up video, "Euclidean Distance: Math, Applications and Code #SoME3," the mathematical foundations of Euclidean distance are discussed alongside its real-world applications and implementation in Python.

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