I can't prove what people 2300 years ago could

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Darya

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So my question is concerning Archimedes' Quadrature of Parabola.
In his theorem I understand everything except for why the area of [MATH] ∆ACD [/MATH] and [MATH]∆CEB[/MATH] is [MATH]1/8×S∆ABC[/MATH]. So I decided to try to prove it.
All I know is that C is a midpoint between A and B. Tangent to C is parallel to AB. I also tried to drop a line from point B, parallel to FC but I'm still stuck.

Any hints will be hugely appreciated!
 

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Darya said:
So my question is concerning Archimedes' Quadrature of Parabola.
In his theorem I understand everything except for why the area of [MATH] ∆ACD [/MATH] and [MATH]∆CEB[/MATH] is [MATH]1/8×S∆ABC[/MATH]. So I decided to try to prove it.
All I know is that C is a midpoint between A and B. Tangent to C is parallel to AB. I also tried to drop a line from point B, parallel to FC but I'm still stuck.

Any hints will be hugely appreciated!
Click to expand...
Please post complete text (or link) of whatever you are referencing.
 
Darya said:
So my question is concerning Archimedes' Quadrature of Parabola.
In his theorem I understand everything except for why the area of [MATH] ∆ACD [/MATH] and [MATH]∆CEB[/MATH] is [MATH]1/8×S∆ABC[/MATH]. So I decided to try to prove it.
All I know is that C is a midpoint between A and B. Tangent to C is parallel to AB. I also tried to drop a line from point B, parallel to FC but I'm still stuck.

Any hints will be hugely appreciated!
Click to expand...
In the Wikipedia page (which of course just summarizes the flow of the proof) says that it took propositions 18 through 21 to prove this fact -- no one said that anyone back then would say it was immediately obvious!

Can you see why what they say about the height and width implies the conclusion about the areas? Or is your issue with proving the fact about height and width?

Did you look at the External Links at the bottom? I sometimes use Wikipedia as an index to knowledge found elsewhere, by focusing on those links rather than the content itself.
 
Dr.Peterson said:
In the Wikipedia page (which of course just summarizes the flow of the proof) says that it took propositions 18 through 21 to prove this fact -- no one said that anyone back then would say it was immediately obvious!

Can you see why what they say about the height and width implies the conclusion about the areas? Or is your issue with proving the fact about height and width?

Did you look at the External Links at the bottom? I sometimes use Wikipedia as an index to knowledge found elsewhere, by focusing on those links rather than the content itself.
Click to expand...
Yes, of course I see that, if the new triangles have half of width and 1/4 of height of the original one, their areas will be 1/8 of the big triangle. I have a hard time understanding how the proportion was found.

I did look at the external links. https://proofwiki.org/wiki/Quadrature_of_Parabola
Here I even found a proof but then I'm struggling to understand the part about EF (attached file). Any hint??
 

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Darya said:
I did look at the external links. https://proofwiki.org/wiki/Quadrature_of_Parabola
Here I even found a proof but then I'm struggling to understand the part about EF (attached file). Any hint??
Click to expand...

I assume you're asking about how they get the first line of that, not how they get each line from the line before, which is just algebraic manipulation.

The first term is the y-coordinate of E; the quantity subtracted from that is the y-coordinate of F, using the equation of the line CQ, with slope [MATH]2ax_1[/MATH].

Of course, this is nothing like Archimedes' proof!
 
Dr.Peterson said:
I assume you're asking about how they get the first line of that, not how they get each line from the line before, which is just algebraic manipulation.

The first term is the y-coordinate of E; the quantity subtracted from that is the y-coordinate of F, using the equation of the line CQ, with slope [MATH]2ax_1[/MATH].

Of course, this is nothing like Archimedes' proof!
Click to expand...
Thanks so much for your replies!

Yes, I guess the bit about F coordinate is the only thing that's not clear to me. Namely, how they got y-coordinate of F by adding y-coordinate of C and slope of CQ times x-coordinate of F (which is the equation of the line CQ, right?)
A bit sloppy in analytical geometry...
The rest of the proof seems fine.

Well, maybe some day by exhaustion I will get to Archimedes level .
 
Darya said:
Well, maybe some day by exhaustion I will get to Archimedes level .
Click to expand...
Don't count on it. Just because he lived long ago doesn't mean he didn't have a towering intellect. He might have invented calculus had he not been killed by a Roman soldier. It likely set mathematics back about 2000 years.
 
Dr.Peterson said:
I assume you're asking about how they get the first line of that, not how they get each line from the line before, which is just algebraic manipulation.

The first term is the y-coordinate of E; the quantity subtracted from that is the y-coordinate of F, using the equation of the line CQ, with slope [MATH]2ax_1[/MATH].

Of course, this is nothing like Archimedes' proof!
Click to expand...
Got it now! Thanks~
 
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