Theorem of Pythagoras

The Pythagorean Theorem which applies to right triangles (an algebraic truth about the sides and hypotenuse) can be proven using the fact that the area of a outer square equals the sum of its interior areas. The basic idea;

is used often in science and engineering.

The image shows a square drawn inside of a larger square. The lengths A, B, C, and the acute angle θ are identified.

Prove  the Theorem of Pythagoras.
♦  By the sketch, we see that the area of the circumscribed square equals the four areas of inscribed triangles plus the area of the inscribed square:

(A + B)² = 4 (AB/2) + C²

By a previous proof we know: (A + B)² = A² + 2AB + B². With this substitution we have:

A² + 2AB + B² = C² + 4 (AB/2)

A touch of algebra and we have the Pythagorean Theorem:

A² + B² = C².  QED.

We can take this result a step further.
♦  First divide the theorem by C², to obtain:

(A/C)² + (B/C)² = 1

Next realize for a right triangle, sin(θ) is defined as A/C and cos(θ) is defined as B/C. Make these substitutions into the theorem to obtain the famous formula:

sin²θ + cos²θ = 1  QED.

Theorem of Pythagoras

The Pythagorean Theorem, a statement of an algebraic truth regarding the sides and hypotenuse of right triangles, can be proved by use of geometry and the general truth:

"The whole equals the sum of its parts."