Basic Proportionality Theorem

Thales Theorem - The Foundation of Similar Triangles

"Thales of Miletus (640-546 BC) - Ancient Greek mathematician who discovered this fundamental relationship between parallel lines and proportional segments"
If DE || BC in triangle ABC, then AD/DB = AE/EC
When a line is parallel to one side of a triangle, it divides the other two sides proportionally
Welcome to the fascinating world of the Basic Proportionality Theorem! Let's discover how parallel lines create perfect proportional relationships.
🏛️ Historical Significance

Thales of Miletus discovered this theorem around 600 BC, making it one of the oldest known mathematical theorems. It forms the foundation for understanding similarity and proportionality in geometry.

Ancient Wisdom
Still fundamental to modern geometry
📐 Basic Proportionality

When a line is drawn parallel to one side of a triangle, it creates two smaller triangles that are similar to the original triangle and to each other.

DE || BC ⟹ AD/DB = AE/EC
Parallel lines create proportional segments
🔄 Converse Theorem

If a line divides two sides of a triangle in the same ratio, then that line must be parallel to the third side. This provides a test for parallelism!

AD/DB = AE/EC ⟹ DE || BC
Equal ratios guarantee parallelism
🏗️ Applications

This theorem is essential for solving problems involving similar triangles, scale drawings, map reading, and architectural designs where proportional relationships matter.

Real-world geometry
From pyramids to skyscrapers

🎮 Interactive Proportion Explorer

Triangle Controls:

Measurements:

AD/DB = 1.00
AE/EC = 1.00
Parallel? No