Expert Guidance for Your Pythagorean Theorem Questions

Similar Triangles and the Pythagorean Theorem

Derive the Pythagorean Theorem by eliminating the x.
I have already shown that $\mathrm{△<!--\; △\; -->}BAE$ and $\mathrm{△<!--\; △\; -->}BDE$ are congruent, and that $\mathrm{△<!--\; △\; -->}EDC$ and $\mathrm{△<!--\; △\; -->}BAC$ are similar triangles. However, I am having trouble setting up the resulting proportions and using that to derive the Pythagorean Theorem. I believe the resulting proportions are
$\frac{BA}{DE}=\frac{AC}{DC}=\frac{BC}{EC}.$.
Any help will be appreciated!

Pythagorean theorem in higher dimensions?
When using vector notation in coordinate systems (Cartesian coordinates) we see that the magnitude of a vector in two dimensions is equal to the square root of its Y component squared added to its X component squared (Pythagorean theorem).But the same calculation is done for a three dimensional vector that has X, Y, and Z components.Is there a triangle that has four sides? (of course not, but how does this right triangle formula work for a calculation that involves more than two dimensions?).

How does the attached figure prove the Pythagorean theorem?

Triangle law of vector addition vs Pythagorean theorem
Suppose there is a vector a of magnitude 5 units to the east, another vector b of magnitude 6 units to the north. To find magnitude of vector a + vector b,
By the triangle law of vector addition, it is 5 + 6 units = 11 units.
By Pythagorean theorem, it is $\sqrt{(}{5}^2+6^2)=\sqrt{(}61)$
Which answer is right? If so, why is the other wrong?
Thank you!

Find the length of base of a triangle without using Pythagorean Theorem
I'm curious whether it is possible to find the length of base of the triangle without using Pythagorean Theorem
No Pythagorean Theorem mean:
=> No trigonometric because trigonometric is built on top of Pythagorean Theorem. etc $\sin <!--\; \; -->\mathrm{\theta <!--\; \theta \; -->}=\frac{a}{r}$
=> No Integration on line or curve because the integration is built on top of Pythagorean Theorem. etc: $s(x)=\int <!--\; \int \; -->\sqrt{f^{\prime }(x{)}^2+1}$

How did Pythagoras argue for the Pythagorean theorem?
There are lots of proofs for the Pythagorean theorem. But, how did Pyhtagoras argue for the theorem himself?

How to state Pythagorean theorem in a neutral synthetic geometry?
In some lists of statements equivalent to the parallel postulate (such as Which statements are equivalent to the parallel postulate?), one can find the Pythagorean theorem. To prove this equivalence one has first to state the pythagorean theorem in neutral geometry (I name 'neutral geometry' a geometry in which parallel lines do exist but with the parallel postulate removed).
If one start with an axiom system like Birkhoff's postulates which assume reals numbers and ruler and protractor from the beginning then there is no problem stating the Pythagorean theorem.
My question is how one can one state the Pythagorean theorem in a neutral synthetic geometry based on axioms such as Hilbert's axioms group I II III or Tarski's axioms A1−A9 ?
It is possible to define segment length in neutral Tarski's or Hilbert's geometries as an equivalence class using the congruence (≡) relation. It is also possible to define the congruence of triangles.
However, the geometric definition of multiplication as given by Hilbert assume the parallel postulate. The existence of a square is equivalent to the parallel postulate.

which axiom(s) are behind the Pythagorean Theorem
There are many elementary proofs for the Pythagorean Theorem, but no matter they use areas, similarities, even algebraic proofs, it is not straightforward to tell why it is true tracing back to the (Euclidean geometry) axioms. Are all these proofs equivalent? Do they all track back to the same axioms?

Pythagorean theorem does require the Cauchy-Schwarz inequality?
In Wikipedia I found a proof of Schwarz inequality making use of the Pythagorean Theorem. Anyway I'm wondering... isn't that a petitio principii? Isn't the Pythagorean Theorem in Hilbert spaces proved by the Cauchy-Schwarz inequality? Or effectively is the Pythagorean Theorem a stronger if not equivalent notion?

Help with Pythagorean theorem proof for Hilbert Spaces
For given functions f and g in a Hilbert space $L^2(a,b)$, prove if the Pythagorean theorem is true for f and g, then it is also true for cf and cg, where c is a constant. What would $<cf,cg>$ be?
Assume the Pythagorean Theorem holds for functions f and g in a Hilbert space $L^2(a,b)$. Then $||f|{|}^2+||g|{|}^2=||f+g|{|}^2$.
I have the beginning of the proof above however I am having trouble continuing.