The formula $a^2+b^2 = c^2$ is common knowledge and the words for hypotenuse and leg (is "cathetus" not used in English?) are basic mathematical vocabulary. Including these seems a good idea.
Connections to other mathematics
The notation with AB, CA and BC might be something the students have used or will use in less analytical ...
Two alternatives I have seen used (am not necessarily recommending them, but will list some pros and cons) which don't seem to have been mentioned yet.
Don't denote it algebraically at all! Draw a picture instead
For the lower end of the 12-16 age range, I've seen this work really well. You literally draw the squares sticking out from the triangle. Write the ...
In Olympiad geometry, $a$, $b$, $c$ is the so-called standard notation for the sides of a triangle, so it makes sense to use it consistently when referring to a triangle (in isolation). However, in any case the general principle is introducing all your notation. Writing
Pythagoras' theorem states that $a^2+b^2=c^2$.
Pythagoras' theorem states that $...
These books are simply reflecting the longstanding and universal usage in physics and engineering, which is that these words can have either meaning, and any ambiguity is normally either resolved by context or unimportant.
You asked a big question. Maybe some of these ideas are worth investigating more deeply. The two first explanations are completely generic, while the last two are more specific to mathematics; I would advise to not forget the power of the generic explanations. But I don't know to what extent these are good explanations or what other good ones there might be. ...
The only one of these that looks objectionable to me is the one that calls the hypotenuse $h$, since in a triangle the letter $h$ usually refers to the triangle's height (which could be either one of the legs but could not be the hypotenuse).
In engineering and physics, in the English literature, angular frequency is oftentimes abbreviated in frequency. Sometimes this is explicitly stated at the beginning, but mostly it is given implicitly. However, the intended audience is expected to not make any confusion. The usage is certainly not recent, and I think I've seen it already about 40 years ago ...
In my (lack of) experience, the second class in Euclidean geometry is actually an undergraduate course that seems to be often called College Geometry. And, yeah, there are so many fascinating topics there that are accessible to gifted HS students that don't really measure up to other topics when it comes to career and college readiness, but they were the ...
A quick Google search returned the following texts based on key words from your posting: rational circle, Euler line, geometry textbook. Based on skimming the contents and prefaces, I think they are good chance to match your desires:
A Beautiful Journey through Olympiad Geometry, Stefan Lozanovski, 2016
I like The Art and Craft of Problem Solving, by Paul Zeitz. But you probably already know of it, since it's mentioned on the aops site.
I think it addresses some of what you list above. But it goes so much deeper too.
I took an undergraduate course in "advanced planar geometry" in preparation for secondary teaching. It was from the text by Isaacs, "Geometry for College Students (Pure and Applied Undergraduate Texts)" and is available for purchase here. There are also less upstanding ways to obtain this text.
It appears to cover all the topics you've ...
This is more of a long comment than an answer.
Some of the other answers here are advocating for factoring rather than case analysis. I just want to point out that proving the theorem that for any two real numbers $ab = 0$ if and only if $a=0$ or $b=0$ requires the same sort of reasoning as @Aeryk advocates in the accepted answer:
In one direction, if we ...
Whatever you choose, make sure to follow some basic rules:
Clearly state the preconditions and make sure they are understood (right triangle in your case).
Clearly state the meaning of the symbols (e.g. which symbols stand for the sides adjacent to the right angle, and which for the third one).
Use the symbols consistently (don't make e.g. the same symbol &...
So some great books on geometry and written for middle/high schoolers are those written by Kiselev and translated to English by Alexander Givental. The textbooks are "Planimetry" and "Stereometry" respectively. Maybe try taking a look in either of those books and see if there is something that suits you?
Here's a link to the first.
@MatthewDaly's mention of The Secrets of Triangles reminded me of
the just released A Cornucopia of Quadrilaterals, which I've been reading.
(Alsina, Claudi, and Roger B. Nelsen. Vol. 55. American Mathematical Soc., 2020.)
For example, for a bicentric quadrilateral (cyclic and tangential) of side lengths $a,b,c,d$ and angles $A,B,C,D$:
a + c &...
A common error I see when teaching function composition is students seeing it as multiplication. Many factors contribute to this, but examples with multiplication in them don't help:
Q: Let f(x) = 2x and g(x) = x+1. What is f(g(x))?
A: f(g(x)) = 2(x+1)
Some students will see this and think that the answer somehow involves multiplying f(x) by g(x).