1, geometric proof

Geometric proof is the most common and intuitive proof method of Pythagorean theorem. The basic idea is to use geometric figures and properties to deduce the relationship between theorems. For example, Pythagorean theorem can be proved by drawing a right triangle and using geometric similarity and the area relationship of the triangle.

2. Algebraic proof

Algebraic proof is to prove Pythagorean theorem by algebraic method. The basic idea is to transform Pythagorean theorem into algebraic equality or identity by introducing variables, algebraic operations and equations. For example, theorems can be proved by algebraic techniques such as sum and difference of squares formula and collocation method.

3. Mathematical induction proof

Mathematical induction is a special proof method, which is suitable for integer sets that meet certain conditions. The basic idea is to prove that the theorem holds for a special integer, and then use inductive hypothesis and recursive relation to prove that the theorem holds for all integers that meet the conditions. In the proof of Pythagorean theorem, right triangles with different side lengths can be proved to satisfy the theorem by mathematical induction.

Expand knowledge:

Euclid proof: The proof method of Pythagorean theorem given by Euclid is a geometric proof. Euclid proved the geometric properties of Pythagorean theorem by drawing several right triangles.

Newton proof: Newton proof of Pythagorean theorem is one of algebraic proof methods. He expressed the side length of a right triangle as an algebraic expression, and finally got the equation of Pythagorean theorem through algebraic operation and equation solution.

Riemannian geometry proof: Riemannian geometry is a non-Euclidean geometry theory, and there is a proof method of Pythagorean theorem based on geometric figures. Pythagorean theorem can be proved by drawing an arc on a two-dimensional plane and using the arc length to represent the multiple of the side length of a right triangle.

Pythagorean theorem can be proved by geometric proof, algebraic proof and mathematical induction. Geometric proof is the most intuitive method, algebraic proof is through algebraic operation and equation solving, and mathematical induction is suitable for integer sets. In addition, mathematicians such as Euclid, Newton and Riemannian Geometry gave different proof methods, which enriched the understanding and application of Pythagorean theorem.