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#1 2013-06-03 04:12:48

{7/3}
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Registered: 2013-02-11
Posts: 210

Another geometry proof

This proof isn't actually needed by me,but i am curious how to prove it,"the diameter of a circle divides the circle into 2 equal parts"


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#2 2013-06-03 04:26:24

Agnishom
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Re: Another geometry proof

Define a diameter


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#3 2013-06-03 06:20:56

bob bundy
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Re: Another geometry proof

hi {7/3}

Back to Euclid to see what he made of this.  In book 1 of The Elements he has it as an axiom.

17. And a diameter of the circle is any straight-line, being drawn through the center, and terminated in each
direction by the circumference of the circle. (And) any such (straight-line) also cuts the circle in half.

The second part doesn't actually follow directly from the first so I guess a proof is still needed.  I'll give it some thought.  smile

Bob


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#4 2013-06-03 19:54:42

{7/3}
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Registered: 2013-02-11
Posts: 210

Re: Another geometry proof

Never mind,i've found a proof.


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#5 2013-06-03 23:52:37

Agnishom
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From: The Complex Plane
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Re: Another geometry proof

Please post the proof


'And fun? If maths is fun, then getting a tooth extraction is fun. A viral infection is fun. Rabies shots are fun.'
'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'
'Humanity is still kept intact. It remains within.' -Alokananda

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#6 2013-06-04 03:27:33

{7/3}
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Registered: 2013-02-11
Posts: 210

Re: Another geometry proof

(sorry,can't post pics,somethings wrong with my browser and my english is also shaky)
if n diameters are drawn in a circle than 2n triangles are drawn,n on each side of a specific diameter.for every triangle there is a congruent triangle on other side(2 sides are radii,interior angles are vertical) thus n triangles on one side are congruent to n on the other side.as n tends to infinity the area becomes equal to other sides area.so, both sides have equal area.

Last edited by {7/3} (2013-06-04 03:36:00)


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#7 2013-06-08 14:13:20

Agnishom
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From: The Complex Plane
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Posts: 15,355
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Re: Another geometry proof

That is still not clear. Could you give a link?


'And fun? If maths is fun, then getting a tooth extraction is fun. A viral infection is fun. Rabies shots are fun.'
'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'
'Humanity is still kept intact. It remains within.' -Alokananda

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#8 2013-06-08 17:32:33

{7/3}
Member
Registered: 2013-02-11
Posts: 210

Re: Another geometry proof

Sorry,i didn't find it in the net,i kinda discovered it.


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#9 2013-06-08 20:57:26

anonimnystefy
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Registered: 2011-05-23
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Re: Another geometry proof

Hm, can you prove that it will  tend to the area as n goes to infinity.


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#10 2013-06-08 23:14:01

{7/3}
Member
Registered: 2013-02-11
Posts: 210

Re: Another geometry proof

I discovered another proof,the angle standing on a diameter is right,so the angles standing on both side of diameter are right and equal.and equal angles in a circle are stand on similar amd equal arc (congruent arcs).so,the arcs on both side of diameters are equal and it follows the sides are equal in area.

Last edited by {7/3} (2013-06-08 23:18:32)


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#11 2013-06-09 00:45:51

{7/3}
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Registered: 2013-02-11
Posts: 210

Re: Another geometry proof

anonimnystefy wrote:

Hm, can you prove that it will  tend to the area as n goes to infinity.

my textbook uses the fact that as n tends to infinity sum of the triangles' base equals to circumference.I'd love to prove it but i'm busy cause my exams have started...


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#12 2013-06-24 00:18:13

{7/3}
Member
Registered: 2013-02-11
Posts: 210

Re: Another geometry proof

Okay,my exams have ended,as n tends to infinity the bases will become infinitesimal,it can be proved using law of cosines.and as the n tends to infinity angle will become small and arc will become small,so both arc and base will become equal and it follows that area will be equal

Last edited by {7/3} (2013-06-24 00:22:13)


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