Fun geometry stuff

[KyriakosCH]

I may have more to post in the future, but here is a first:

Already in the ancient era (as Proclus notes in his comments on Euclid) there were a few people who thought construction in geometry as being bypassed when the premise seems to allow you to do things in a single move.
A good example of this would be the second proposition in the first book of Euclid (pic provided - I actually modeled this in Blender  ), where Proclus (5th century byzantine from Constantinople) says that some people just used the compass to secure a radius of BC, then created a circle from A.
But it seems that Euclid's reason for presenting this so early on was exactly to highlight that such a thing wouldn't be a valid construction, and in logic it's known as "begging the question" (petitio principii). This becomes important, since likewise in formal systems (computers and other stuff) you can't just go outside the system and derive a new formula, even if it is obvious outside the system; you have to mention either specific theorems or alter the previous statement by one of the allowed rules. It's the same here: you have to construct BC from something else (despite using a radius BC to get there), and just using the compass to move length BC around would be analogous to examining a statement from outside the system.



(the circles from A and B, with radius AB, are how to construct the equilateral triangle; the other vertex is where those circles intersect- proposition I)

(I also provided an alternative construction, without use of Proposition I but still using radius BC only from a center in known points of BC)

[KyriakosCH]

Ok, to finish off with Proposition II (all unique cases)

Here is the construction in the method of Euclid (that is, with the use of an equilateral triangle), in the case that AB>BC and AB=BC.



The only other different cases are symmetrical (A is to the other side of BC, or is below/above them).
But If A is actually in the segment BC (which likely one can infer isn't the case, from the proposition's wording) then you can just use the same idea as in the alt method I posted, and which you can see in the lower pair of constructions (do notice that in them you never construct the circle with radius equal to BC; I provided a construction with equilateral triangle and one without).

[KyriakosCH]

Ok... Here is the first part (one case) of Proposition III

It should be stressed again that Euclid proceeds always from the previously proven constructions, in a type of procession of theorems or true sentences.
So in Proposition II you use Proposition I (how to construct an equilateral triangle given only a line segment and compass), and in III you use II (how to move a segment so that it now starts from another given point).



(ps: you might notice that I deliberately made it so CD = 1/2 of AB I like it, since now it looks like something Paul Klee would have drawn)

[KyriakosCH]

Cosines (and sines, etc) didn't exist at the time of Euclid. However you can easily arrive at the law of cosines using Proposition 12 (if the triangle has an obtuse angle) or 13 (if it hasn't).

[Mandle]

Math is indoctrination. We don't live in a realm described by formulas.
Globe is dead. Earth is flat.
Space is fake.
RIP globe Earth NASA lies.

[KyriakosCH]

I know, but TPTB don't allow me to look past the lie 

[Mandle]

I know, but TPTB don't allow me to look past the lie 

Hahaha nice comeback. I had to Google TPTB which I guess outs me as not a real conspiracy nut.

[KyriakosCH]

See how easy it is to arrive from tic-tac-toe to Gary Oldman's Dracula?