The Pythagoras Theorem that is not discovered by Pythagoras.

The Pythagorean theorem is a name that resonates within the realm of geometry, likely encountered countless times. However, before delving into the captivating world of this theorem, it is essential to grasp the origins of geometry itself. The term "geometry" finds its roots in the Greek words "geo," meaning Earth, and "metron," meaning measurement. Geometry's inception can be traced back to one of its fundamental aspects: angles. The discovery of angles can be attributed to the astute astrologers of Ancient Babylonia, who sought to measure the angles between stars and the sun. While the journey of geometry's exploration is indeed a fascinating one that warrants thorough examination, let us narrow our focus to Pythagoras' theorem for now.

In the accompanying image, you can observe the earliest form of this famous theorem. Despite the common misconception, it was not Pythagoras who first discovered or originated this theorem. Its origins can be traced back to 1900 B.C. when ancient Egyptians and Babylonians utilized it in constructing pyramids and other auspicious structures. However, their understanding of the theorem was not as refined as that developed by Pythagoras, who recognized its significance in 450 B.C. Pythagoras was the first to provide a formal proof—a deductive geometrical proof—of the theorem.

Two proofs can be explored: the Pythagorean proof, which is fundamental and impressive, and the proof discovered by us, which leverages trigonometry or similar triangles. The accompanying image offers a basic idea of how to prove it, but in case you find it challenging, consider the areas of triangles and squares in the figure. I invite you to share images of your own proofs in the comment box.

Consider a right-angled triangle with sides of lengths 'a', 'b', and 'c', where 'c' is the hypotenuse—the side opposite the right angle.

We start by drawing a perpendicular line (altitude) from the right angle to the hypotenuse. This line divides the triangle into two smaller triangles.

Let's label the length of the altitude as 'h', and it divides the hypotenuse 'c' into two segments of lengths 'x' and 'y'.

Now, let's consider the two smaller triangles formed: the first triangle with sides 'a', 'h', and 'x', and the second triangle with sides 'b', 'h', and 'y'.

By considering the similar triangles formed by the altitude, we can establish the following ratios:

In the first smaller triangle:

a / x = h / a

In the second smaller triangle:

b / y = h / b

Cross-multiplying these ratios, we get:

a^2 = xh [from the first triangle]

b^2 = yh [from the second triangle]

Adding these two equations together, we obtain:

a^2 + b^2 = xh + yh

Rearranging, we have:

a^2 + b^2 = (x + y)h

Since x + y is equal to the length of the hypotenuse 'c', we can substitute c for x + y in the equation:

a^2 + b^2 = c * h

Now, by examining the right-angled triangle, we can see that the altitude 'h' is the height of the triangle.

Using the formula for the area of a triangle, which is equal to 1/2 * base * height, we have:

1/2 * a * b = 1/2 * c * h

Simplifying, we get:

ab = ch

Substituting this into our previous equation, we have:

a^2 + b^2 = ab

Now, subtracting ab from both sides, we obtain:

a^2 + b^2 - ab = 0

Factoring the left side of the equation, we have:

(a^2 - 2ab + b^2) - ab = 0

(a - b)^2 - ab = 0

Expanding further, we get:

a^2 - 2ab + b^2 - ab = 0

a^2 - ab + b^2 = 0

Now, adding 2ab to both sides, we obtain:

a^2 + 2ab + b^2 = 2ab

(a + b)^2 = 2ab

Dividing both sides by 2, we have:

(a + b)^2 / 2 = ab

Taking the square root of both sides, we get:

(a + b) / √2 = √ab

Multiplying both sides by √2, we obtain:

(a + b) = √2 * √ab

Simplifying, we have:

a + b = √2 * √ab

Squaring both sides, we get:

(a + b)^2 = 2ab

Expanding the left side, we have:

a^2 + 2ab + b^2 = 2ab

Finally, simplifying the equation, we arrive at:

a^2 + b^2 = c^2

Thus, the proof is complete. The equation a^2 + b^2 = c^2 represents the Pythagorean theorem, stating that the square of the hypotenuse (c^2) is equal to the sum of the squares of the other two sides (a^2 and b^2) in a right-angled triangle.

The Pythagorean theorem finds applications in various areas, including:

1. Determining the lengths of sides in right-angled triangles.

2. Calculating the length of the diagonal in rectangles, squares, and other shapes.

3. Utilizing trigonometry to find ratios such as sine, cosine, tangent, cosecant, secant, and cotangent.

4. Employing it in security cameras for face recognition.

5. Utilizing the technique in engineering and construction by architects.

6. Applying it in surveying mountains.

7. Utilizing it in navigation to find the shortest route.

8. Deriving formulas for base, perpendicular, and hypotenuse measurements.

9. Assisting painters in utilizing ladders to paint high buildings.

10. Calculating the steepness of slopes on hills or mountains.

11. Utilizing the converse of the Pythagorean theorem to determine if a triangle is a right triangle or not.