Can someone please explain to me why DE would necessarily be equal to EB?This is from Galileo Galilei of all people, for fucks sake. This is a huge hole for me in his authorship, other than him composing a line out of points for some reason, even though Euclid's first definition exists. (also he should have used Euclid III.36 to define where that line was tangent to the circle, as AC, AG should be equal to the square on AE)If no one is able to help me, it wouldn't surprise me, as no one actually knows anything about actual physics or astrophysics on here.
I don't know anything about physics or astrophysics but I like what you are doing
>>16536814Respect, sir. If you read classic scientific literature, that actually comes in handy now and then.
>in which OP watches Galileo bisect a kite, and proceeds to get angry
>>16536795AD is tangent to the circle centered at C, so ADC is 90 degrees. EB is perpendicular to CF, so EBC is 90 degrees. CDE and CBE share the side opposite the 90 degree angle. This forces DC=CB and DE=BE, otherwise it would be impossible for CE to share a side or for CDE=CBE=90 degrees. The construction forces both conditions, so both triangles are not only similar, but mirror images along CE.
>>16536844After thinking about it for more than two seconds, you are correct. However, the way for Galileo to prove it, would have been to state that it is a right angled triangle and that DC, CE = BC, CE. Has nothing to do with sharing a side, but in either case, we both proved it better than Galileo. Always surprises me sometimes how mathematicians of antiquity don't do things perfectly sometimes. I preferred Johannes Kepler to Galileo Galilei for this very reason.
>>16536844Actually you're not right, you need to know that two sides are equal to make the triangles congruent. Just because the hypotenuse is equal and shared in a 90 degree triangle does not mean that the triangles are congruent, however knowing that two sides are congruent and the 90 degree angle does. So Galileo did not use the correct proof. He should have stated that DC was equal to BC, something known. Instead he stated DE was equal to BE, something not proven within that proof. Either way, the entire proof follows, so nothing is fundamentally taken away, except my respect for Galileo a bit.
>>16536914When he made circle center c that made DC and bc radii of circle center c and so equal?
>>16536795Are you asking that given CD = CB (circle radius), and the hypotenuse of the right triangles are the same, you can't figure out if the 3rd side of each is equal? Feel like the Pythagorean Theorem is the straight answer
>>16536954Debc would be the kite that is bisected by ec
>>16536967Correct, so Galileo did not prove this correctly. He went about this with the wrong sides.
Euclid doesn't mess with kites
>>16536795>Can someone please explain to me why DE would necessarily be equal to EB?Notice the equality of angles BEF=EAF=DAC. Call this angle x. Use a unit circle for simplicity.sin x = FB = 1/FA = DC/CAcos x = BE = EA/FA = DA/CADE = EA - DA = (FA-CA) cos x = FC cos x = 1 * cos x = BE
>>16537116what are you talking about? I read the book, he did everything fine. He obviously knows about the pythagorean theorem, and the two triangle are clearly the same so the angles are the same etc etc etc
continuing >>16537869, Side-Side-Angle is already enough to make a unique triangle by the law of cosines, idk what you're confused about.
>>16537869I read the page* you posted
>>16537870Law of cosines and Galileo don't go together, you can't retroactively give him law of cosines as you understand it
Also euclid doesn't use side side angle and that's interesting and makes me wonder if Galileo would have avoided it as well
>>16537888>>16537893Well he definitely knew if for a right triangle; the pythagorean theorem - a special case of the law of cosines - was well known. Takes like 10 seconds to figure out they're equal, almost trivial.
continuing >>16537986, it also looks like Euclid's book had the law of cosines. Wouldn't bet against them understanding side-side-angle, especially when even without rigour it's obviously true to any layman, which would mean geometers definitely sought out a proof for it.
I definitely recall learning SSA in geometry class actually. Was like the last of the SAS,SSS etc though
>>16537986Euclid doesn't mention law of cosines or Pythagoras, it's simply called proposition 47
>>16538223Who cares if you call it Pythagorean theorem or Plimpton's theorem or Einstein's theorem? What kinda useless argument is that? And he does have a law of cosines. Again, you're making zero argument; Galileo does nothing wrong here.
>>16537990To be fair, the diagram in question was constructed with compass and straightedge. There's not even any rules required when you can measure.
>>16538538You are arguing about a book you never read you absolute silly goose
>>16538563Correct
Why do you never see these diagrams in public school? I feel like people should develop an intuition for them. They are pretty cool.
>>16540618I don't like your diagram
>>16540719Its not mine but why?
>>16540776The language of it puts me off, that's all
