Greetings, dear Readers! Today I want to talk about the recently discovered amazing proof of the Pythagorean theorem. Yes, yes, you heard right!
Even in this, it would seem, up and down the plowed field, an uncultivated strip remained. Here’s the thing: it was believed that any proof of the Pythagorean theorem through trigonometric functions somehow boils down to the application of the basic trigonometric identity:
, which in itself is one of the ways to write the theorem. Thus, using this expression, we “will go out on ourselves”, i.e. to prove what we accept as truth at the very beginning.
Fragment from the book with the largest known collection of proofs of the theorem – “The Pythagorean Proposition” Elisha Loomisa Loomisa. Quote: “Looking ahead, the thoughtful reader may wonder: Are there proofs based on trigonometry or analytic geometry? Trigonometric proofs do not exist because: all the fundamental formulas of trigonometry are based on the truth of the Pythagorean theorem; based on this theorem, we say sin^A + cos^A = 1, etc. Trigonometry exists thanks to the Pythagorean theorem
However recently a proof of the Pythagorean theorem was found, based on another fundamental statement – the sine theorem for a triangle. So, consider an isosceles triangle (it is in it that the bisector is the height, and this is important to us):
Let’s write the sine theorem for this triangle:
Now we need some geometric constructions and a written ratio of the sides of similar rectangles:
Now we continue to apply similarity properties for each subsequent right triangles:
That is, for each pair:
Write down the similarity relation
We express the common leg through the sine of the angle
Substitute and get the new segment length
As you can see, it turns out something wonderful. There are two geometric progressions on the face:
And now let’s carefully see that the two sides we found are the hypotenuse and the leg of a right-angled triangle with an angle of 2 * alpha:
In the end, we returned to the formula we derived from the very beginning and “discovered” the Pythagorean theorem!
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