imo the reason most students find this proof difficult would be that they haven't figured out the subtraction law of vectors yet ( I had been one of those students)

$\Delta t=\text{distance}(1,2)/v=2r\phi/\Delta t$ The final $\Delta t$ should be a $v$.

@G.Smith I'm agree totally with you. Now I improved my question and as usual there is a request of close it. Why?

Whoever voted to close (not me) thinks it is opinion-based. I think your proof is fine for students who don’t know calculus. It seems very clear to me. I prefer your proof to any of the current answers.

@G.Smith Just mine it was a curiosity and I have written with simple words hoping that it was clear. have serious problems: they, also, not remember the most simple definitions: for example of when a motion is said circular and uniform. Although I make infinite efforts they always want high votes and study very little. :-(

Duplicate?

I was pointing out that @MichaelSeifert 's answer is essential the same as dmckee --- ex-moderator kitten's answer.

I'm asking a favour. How did you draw that very nice diagram? Would I be able to instal the software that you used on my Mac computer? [I know almost nothing about IT.]

Personally, I think it's a mistake to start with the math, to require that understanding, and then assume that the intuition is obvious. Most people never get past the math because it's not connected to anything, myself included. But if you start with the intuition, and completely forget the math until the intuition is there, then the same math becomes almost trivial.

@AaronD I have explained the vectorial notations and many exercises for two months. I have given to my students also the videos and illustrations. I accept your comment but it is not possible, IMHO, that all can be very simple. It is important to study like many years ago.

@PhilipWood Ahhahahah..hi, You not must install nothing! :-)))) You use this editor...that I use also for LaTeX :-)...I am not wizard but I think that in this world it is important to leave a great gift before to dead. mathcha.io/editor If you have occurred of my help I am happy to help you and other users. The image was taken from my textbook...now I put that with Mathcha.

Thank you for answering my question about your diagram. What I really want to do is to draw my own diagrams. I used to use an excellent application called 'Freehand', which had Bezier tools for curve-drawing, and lots of other useful stuff, but the application is no longer supported. But I'll look more closely at Mathcha.

@PhilipWood IMHO 'Freehand' it is an obsolete software. I know a bit of graphic design and and is currently used Adobe Illustrator, QuarkXPress. Also Mathcha has Bezier tools for curve-drawing, and lots of other useful :-) and you can export also in TiKZ/LaTeX.

Thank you. I know that Freehand is obsolete, but it did exactly what I wanted. I must investigate mathcha more thoroughly. Can I export stuff done in Mathcha to an ordinary Word document? As I said, I know almost nothing about IT.

@PhilipWood If you want to create images in .png, .pdf, .svg, into a word document etc the answer is yes.

Thank you. I shall persevere.

@PhilipWood : are you familiar with GeoGebra?

@downvoter: What is the reason of your downvotes to the question and answers? Thank you for you explanation.

No I'm not familiar with GeoGebra, but I have heard of it!

@PhilipWood it's a lovely tool with the sky as the limit. Check out these illustrations for example. And next time, please mention the person you are answering to (if you want them to be notified of your answer).

@polfosol. Thank you for your comment and example. I was answering Sebastiano, who – as the comment was under his question – would surely be automatically informed of my comment in his inbox.