How can I prove the following theorem

Establish and prove

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Learn how to prove the Pythagorean Theorem, named after Pythagoras of Samos (* around 570 BC; † after 510 BC). It was known long before Pythagoras, and the Babylonians and Egyptians lived as early as 1600 BC. Recognized the connections in the right triangle and took them for granted.

Establish and prove

A proof is a logical justification with general validity. For example, if you want to prove the Pythagorean theorem, it is not enough to check the equation on a few right-angled triangles. The justification “No counterexample has yet been found” is also insufficient. The validity of the theorem has to be proven for all right triangles, only then is it a mathematical proof. There are hundreds of proofs of the Pythagorean Theorem, some of which differ very little from each other. That is why long-known evidence is constantly being rediscovered. One of these proofs is attributed to Leonardo da Vinci and the American President James Abraham Garfield has also provided proof.
Supplementary evidence This evidence probably also has its origins in ancient India. The idea is to add four identical triangles to the hypotenuse square on the one hand and the two cathetus squares on the other so that two squares of the same size are created. The right triangle with the legs a and b and the hypotenuse c (starting triangle) is given. You look at these two figures. Figure 1 is created by adding four triangles to the hypotenuse square that are congruent to the starting triangle. Figure 2 is created by adding four triangles to the two leg squares that are congruent to the starting triangle. In both figures, blue and orange areas complement each other to form the same square with side length a + b. Since the orange area (four congruent triangles) is the same size in both figures, the blue area is also the same size in both figures. So the two leg squares together are the same size as the hypotenuse square. This only works if the starting triangle is right-angled. If it is not right-angled, the blue and orange surfaces do not complement each other to form a square, but rather to form an octagon. Why is there a corner at point A? The following applies in the starting triangle. It follows . So the right angle at point A together with the angles and do not result in a straight angle. A corner is created.
Leonie and Paul use the same right-angled starting triangle ABC with the side lengths,,. Leonie doubles all sides of the exit triangle. Paul, on the other hand, extends the sides of the exit triangle. Are the two new triangles right angled? Leonie's new triangle is right-angled, Paul's triangle is not right-angled. Reason: Leonie doubles all sides of the initial triangle and receives a new triangle with side lengths a ’=, b’ = and c ’=. You check that the equation for these values ​​is true: So: (By the way: You can justify that Leonie's new triangle A'B'C 'is right-angled with the similarity theorem S: S: S. The triangle A'B'C' arises from the triangle ABC by doubling the sides, i.e. the two triangles are similar to each other. In particular, triangle A'B'C 'has the same angles as triangle ABC and is therefore also right-angled.) Paul extends all sides of the starting triangle by one centimeter and receives a new triangle with the side lengths,,. If this triangle were right-angled, only the longest side c ’would come into question as a hypotenuse. In addition, according to the Pythagorean theorem, the equation should be fulfilled so: ≠