Journey through ancient mathematical wisdom and discover the elegant proof of one of geometry's most fundamental theorems.
The theorem that changed the world of mathematics
Pythagoras (c. 570-495 BCE): Greek mathematician and philosopher
While the theorem bears his name, evidence suggests it was known to ancient Babylonians and Egyptians centuries earlier.
Practical Applications:
Foundation of Geometry:
The theorem is fundamental to understanding distance, coordinate geometry, and trigonometric relationships.
Real-world Applications:
The fundamental relationship in right triangles
Right Triangle with sides a, b, and hypotenuse c
a² + b² = c²
In a right triangle, the square of the hypotenuse equals the sum of the squares of the other two sides.
Key Points:
Multiple ways to prove this fundamental theorem
Step 1: Create squares on each side
Draw squares with areas a², b², and c² on the three sides of the right triangle.
Step 2: Rearrange the pieces
The two smaller squares can be rearranged to exactly fill the larger square.
Conclusion:
Since the areas are equal, a² + b² = c²
Step 1: Start with four identical triangles
Arrange four right triangles with legs a and b to form a large square.
Step 2: Calculate total area
Large square area = (a + b)² = a² + 2ab + b²
Step 3: Alternative calculation
Same area = 4 × (½ab) + c² = 2ab + c²
Conclusion:
a² + 2ab + b² = 2ab + c² → a² + b² = c²
The "Gou Gu" theorem from ancient China (c. 1000 BCE)
Zhou Bi Suan Jing: "The Arithmetical Classic of the Gnomon and the Circular Paths of Heaven"
One of the oldest Chinese mathematical texts, dating back to the Zhou dynasty.
Gou Gu Theorem:
"Gou" (leg) and "Gu" (leg) refer to the two shorter sides of a right triangle.
Practical Use:
Used for astronomical calculations, land surveying, and construction projects.
Step 1: Create a square with side (a + b)
Place four right triangles with legs a and b in the corners.
Step 2: The inner space forms a square
The remaining area in the center is a square with side length c.
Step 3: Calculate areas
Total area = (a + b)² = 4 × (½ab) + c²
Result:
Simplifying: a² + 2ab + b² = 2ab + c² → a² + b² = c²
Explore different proof methods and extensions of the Pythagorean theorem
Method: Using similar triangles
Create similar triangles within the right triangle
Steps:
1. Draw altitude from right angle to hypotenuse
2. Create two smaller similar triangles
3. Use similarity ratios to derive the theorem
Key Insight:
The altitude creates geometric mean relationships
Method: Using vector dot products
Represent triangle sides as vectors
Key Formula:
|a + b|² = |a|² + |b|² + 2(a · b)
For Right Angles:
When a · b = 0, we get |a + b|² = |a|² + |b|²
Special sets of integers that satisfy the Pythagorean theorem
Definition:
Triples with no common factors
Examples:
(3, 4, 5), (5, 12, 13), (7, 24, 25)
Euclid's Formula:
a = m² - n², b = 2mn, c = m² + n²
Conditions:
m > n > 0, gcd(m,n) = 1, m+n odd
Construction:
Creating right angles
Navigation:
Distance calculations
Apply your understanding of the theorem
Given: A right triangle with legs of length 3 and 4 units.
Find the length of the hypotenuse.
Solution:
Using a² + b² = c²:
3² + 4² = c²
9 + 16 = c²
25 = c²
c = √25 = 5
Answer: The hypotenuse is 5 units long.
Given: A right triangle with one leg of 5 units and hypotenuse of 13 units.
Find the length of the other leg.
Solution:
Using a² + b² = c²:
5² + b² = 13²
25 + b² = 169
b² = 169 - 25 = 144
b = √144 = 12
Answer: The other leg is 12 units long.