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

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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Define a diameter

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**bob bundy****Moderator**- Registered: 2010-06-20
- Posts: 7,006

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.

Bob

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

Never mind,i've found a proof.

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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'

'But our love is like the wind. I can't see it but I can feel it.' -A Walk to remember

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

(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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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'

'But our love is like the wind. I can't see it but I can feel it.' -A Walk to remember

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

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

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**anonimnystefy****Real Member**- From: The Foundation
- Registered: 2011-05-23
- Posts: 15,922

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

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

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

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

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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