There is an idea that has been on my mind for a while, and I would like to share it so that it turns into a snowball. Perhaps it will be useful and attractive to engineering enthusiasts. I ask you to help me develop this model that I am thinking of...
Most geometric theorems are proven by putting several theorems together, but there are one or two theorems that represent the core idea of the proof.
If you know the basic theorem you will use to do your proof, you still have to do some work to arrive at the proof, even though the basic step has already been done; Making a very large mind map would be impractical, but small mind maps can be made to express some conclusions
A lot of written mind maps can be done, but I think it would be fun and useful for geometry enthusiasts to have pictorial mind maps so they can guess a good visualization of the proof.
I think that pictorial mind maps are an excellent idea for engineering, as the engineering student has a good connection between the engineering content, and engineering mind maps still carry some challenge as they require work to complete the proof.
Most geometry enthusiasts can understand the statements and requirements of the theorem just from the picture, which is why Arseny Akopyan's book is so popular
I'll post some examples of mind maps I like and I hope my list of answers will expand...
The ground truth in the previous map is the one shown in the upper image; It may be intuitive, and the arrows point to conclusions, not entirely trivial
It is worth noting that the details of the proof need to be done; This is different from what is known as proof without words; If I wanted to turn the proof of the theorem on the right into a proof without words, it would be something like this:
This makes these mental diagrams a narrative of the theorem(s) that can be used in proof; But it still carries the challenge of actually doing this proof
Let's take a second example
If an engineering theorem can be used to prove theorems and perform geometric constructions, a ruler and compass can be placed next to the geometric construction to distinguish it, as in the previous picture.
It can be done the other way around, with one conclusion and many different ways to reach it.
It may be possible to reach one conclusion in many different ways
Double arrows can be used if the inference can be made in both directions, as is the case, for example, with the previous image.
However, there must be a well-understood stereotype for both the premises and the conclusions involved in mind maps of this type.
I would be happy to expand my list with more mind maps. Are there any fans of this idea?
Edit: What I'm looking for in this particular question is more examples of geometry mental diagrams like mine, in which each arrow points to an actual geometric proof