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 ...
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 $...
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 ...
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).
I really wish people would stop teaching the Pythagorean Theorem as $a^2 + b^2 = c^2$, for the following reason: Give your students the diagram below, and ask them to solve for $c$. At least 1/3 of a typical high school class will write $a^2 + b^2 = c^2$ and report back to you that $c = 5$. The problem is that the equation $a^2 + b^2 = c^2$ is so ...
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 &...
I'd say yes, and I'd go with binary if you had to do any one alternative base simply because it's so relevant to computing and technology, and in my experience teaching discrete math, once you understand binary, related bases like octal and hex are pretty simple to pick up. But I don't think the converse is necessarily true.
Ideally I'd like math and CS ...
I can't answer the OP's questions, but I'll just mention that
a local 6th-grade teacher (in the U.S.) has a successful unit on base-$5$.
It is mentioned in the recent article below. Sometimes he called it "star-fish math."
"Math for Grades 1 to 5 Should Be Art."
Mathematical Intelligencer. 42, pages 64–69, Dec. 2020.