Center of a Circle (Diameter)












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I was helping a friend out with an SAT problem, when an odd thought popped up in my head. Why is the center of the diameter of a circle the actual center of the circle? How could I prove this? Is this just the way a circle is defined?






geometry







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    $begingroup$


    I was helping a friend out with an SAT problem, when an odd thought popped up in my head. Why is the center of the diameter of a circle the actual center of the circle? How could I prove this? Is this just the way a circle is defined?






    geometry







    share|cite|improve this question









    $endgroup$















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      5








      5


      0



      $begingroup$


      I was helping a friend out with an SAT problem, when an odd thought popped up in my head. Why is the center of the diameter of a circle the actual center of the circle? How could I prove this? Is this just the way a circle is defined?






      geometry







      share|cite|improve this question









      $endgroup$




      I was helping a friend out with an SAT problem, when an odd thought popped up in my head. Why is the center of the diameter of a circle the actual center of the circle? How could I prove this? Is this just the way a circle is defined?






      geometry




      geometry






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      asked Feb 11 at 4:56









      Dude156Dude156

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          1 Answer
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          $begingroup$

          Great question - a fact which is "obvious" but takes a bit of thought to see for sure why it's true.



          The main thing is to be absolutely clear about what terms mean.



          Definition. A circle (in a given plane) is the set of all points at a given (positive) distance from a given point. The given point is called the centre of the circle. A diameter of the circle is a line segment passing through the centre and having its endpoints on the circle.



          Note that the definition of "diameter" doesn't say that the centre of the circle is its midpoint. It just takes a little more work to prove this.



          So, consider a diameter $AB$. By definition the centre $C$ is somewhere on $AB$. Since $A$ and $B$ are on the circle, the definition says that $AC=BC$. Hence $C$ is the midpoint of $AB$.



          Comment. Euclid's definition (from proofwiki):


          a diameter of the circle is any straight line drawn through the center and terminated in both directions by the circumference of the circle, and such a straight line also bisects the center.




          So it looks as if Euclid just assumed the centre of the diameter is the centre of the circle, and did not even think to ask the question you have asked!!!!






          share|cite|improve this answer











          $endgroup$













          • $begingroup$
            Ahhh, so the center of the diameter is not the center of the circle, but rather the center of the circle is the center of the diameter
            $endgroup$
            – Dude156
            Feb 11 at 5:18








          • 1




            $begingroup$
            No, both of those are true. Just added an extra comment in my answer BTW which I think may interest you :)
            $endgroup$
            – David
            Feb 11 at 5:20










          • $begingroup$
            Ah yes, ole' Euclid was a sad guy... He also assumed distributive property... ;)
            $endgroup$
            – Dude156
            Feb 11 at 5:27












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          1 Answer
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          1 Answer
          1






          active

          oldest

          votes









          active

          oldest

          votes






          active

          oldest

          votes









          10












          $begingroup$

          Great question - a fact which is "obvious" but takes a bit of thought to see for sure why it's true.



          The main thing is to be absolutely clear about what terms mean.



          Definition. A circle (in a given plane) is the set of all points at a given (positive) distance from a given point. The given point is called the centre of the circle. A diameter of the circle is a line segment passing through the centre and having its endpoints on the circle.



          Note that the definition of "diameter" doesn't say that the centre of the circle is its midpoint. It just takes a little more work to prove this.



          So, consider a diameter $AB$. By definition the centre $C$ is somewhere on $AB$. Since $A$ and $B$ are on the circle, the definition says that $AC=BC$. Hence $C$ is the midpoint of $AB$.



          Comment. Euclid's definition (from proofwiki):


          a diameter of the circle is any straight line drawn through the center and terminated in both directions by the circumference of the circle, and such a straight line also bisects the center.




          So it looks as if Euclid just assumed the centre of the diameter is the centre of the circle, and did not even think to ask the question you have asked!!!!






          share|cite|improve this answer











          $endgroup$













          • $begingroup$
            Ahhh, so the center of the diameter is not the center of the circle, but rather the center of the circle is the center of the diameter
            $endgroup$
            – Dude156
            Feb 11 at 5:18








          • 1




            $begingroup$
            No, both of those are true. Just added an extra comment in my answer BTW which I think may interest you :)
            $endgroup$
            – David
            Feb 11 at 5:20










          • $begingroup$
            Ah yes, ole' Euclid was a sad guy... He also assumed distributive property... ;)
            $endgroup$
            – Dude156
            Feb 11 at 5:27
















          10












          $begingroup$

          Great question - a fact which is "obvious" but takes a bit of thought to see for sure why it's true.



          The main thing is to be absolutely clear about what terms mean.



          Definition. A circle (in a given plane) is the set of all points at a given (positive) distance from a given point. The given point is called the centre of the circle. A diameter of the circle is a line segment passing through the centre and having its endpoints on the circle.



          Note that the definition of "diameter" doesn't say that the centre of the circle is its midpoint. It just takes a little more work to prove this.



          So, consider a diameter $AB$. By definition the centre $C$ is somewhere on $AB$. Since $A$ and $B$ are on the circle, the definition says that $AC=BC$. Hence $C$ is the midpoint of $AB$.



          Comment. Euclid's definition (from proofwiki):


          a diameter of the circle is any straight line drawn through the center and terminated in both directions by the circumference of the circle, and such a straight line also bisects the center.




          So it looks as if Euclid just assumed the centre of the diameter is the centre of the circle, and did not even think to ask the question you have asked!!!!






          share|cite|improve this answer











          $endgroup$













          • $begingroup$
            Ahhh, so the center of the diameter is not the center of the circle, but rather the center of the circle is the center of the diameter
            $endgroup$
            – Dude156
            Feb 11 at 5:18








          • 1




            $begingroup$
            No, both of those are true. Just added an extra comment in my answer BTW which I think may interest you :)
            $endgroup$
            – David
            Feb 11 at 5:20










          • $begingroup$
            Ah yes, ole' Euclid was a sad guy... He also assumed distributive property... ;)
            $endgroup$
            – Dude156
            Feb 11 at 5:27














          10












          10








          10





          $begingroup$

          Great question - a fact which is "obvious" but takes a bit of thought to see for sure why it's true.



          The main thing is to be absolutely clear about what terms mean.



          Definition. A circle (in a given plane) is the set of all points at a given (positive) distance from a given point. The given point is called the centre of the circle. A diameter of the circle is a line segment passing through the centre and having its endpoints on the circle.



          Note that the definition of "diameter" doesn't say that the centre of the circle is its midpoint. It just takes a little more work to prove this.



          So, consider a diameter $AB$. By definition the centre $C$ is somewhere on $AB$. Since $A$ and $B$ are on the circle, the definition says that $AC=BC$. Hence $C$ is the midpoint of $AB$.



          Comment. Euclid's definition (from proofwiki):


          a diameter of the circle is any straight line drawn through the center and terminated in both directions by the circumference of the circle, and such a straight line also bisects the center.




          So it looks as if Euclid just assumed the centre of the diameter is the centre of the circle, and did not even think to ask the question you have asked!!!!






          share|cite|improve this answer











          $endgroup$



          Great question - a fact which is "obvious" but takes a bit of thought to see for sure why it's true.



          The main thing is to be absolutely clear about what terms mean.



          Definition. A circle (in a given plane) is the set of all points at a given (positive) distance from a given point. The given point is called the centre of the circle. A diameter of the circle is a line segment passing through the centre and having its endpoints on the circle.



          Note that the definition of "diameter" doesn't say that the centre of the circle is its midpoint. It just takes a little more work to prove this.



          So, consider a diameter $AB$. By definition the centre $C$ is somewhere on $AB$. Since $A$ and $B$ are on the circle, the definition says that $AC=BC$. Hence $C$ is the midpoint of $AB$.



          Comment. Euclid's definition (from proofwiki):


          a diameter of the circle is any straight line drawn through the center and terminated in both directions by the circumference of the circle, and such a straight line also bisects the center.




          So it looks as if Euclid just assumed the centre of the diameter is the centre of the circle, and did not even think to ask the question you have asked!!!!







          share|cite|improve this answer














          share|cite|improve this answer



          share|cite|improve this answer








          edited Feb 11 at 5:20

























          answered Feb 11 at 5:15









          DavidDavid

          69.9k668131




          69.9k668131












          • $begingroup$
            Ahhh, so the center of the diameter is not the center of the circle, but rather the center of the circle is the center of the diameter
            $endgroup$
            – Dude156
            Feb 11 at 5:18








          • 1




            $begingroup$
            No, both of those are true. Just added an extra comment in my answer BTW which I think may interest you :)
            $endgroup$
            – David
            Feb 11 at 5:20










          • $begingroup$
            Ah yes, ole' Euclid was a sad guy... He also assumed distributive property... ;)
            $endgroup$
            – Dude156
            Feb 11 at 5:27


















          • $begingroup$
            Ahhh, so the center of the diameter is not the center of the circle, but rather the center of the circle is the center of the diameter
            $endgroup$
            – Dude156
            Feb 11 at 5:18








          • 1




            $begingroup$
            No, both of those are true. Just added an extra comment in my answer BTW which I think may interest you :)
            $endgroup$
            – David
            Feb 11 at 5:20










          • $begingroup$
            Ah yes, ole' Euclid was a sad guy... He also assumed distributive property... ;)
            $endgroup$
            – Dude156
            Feb 11 at 5:27
















          $begingroup$
          Ahhh, so the center of the diameter is not the center of the circle, but rather the center of the circle is the center of the diameter
          $endgroup$
          – Dude156
          Feb 11 at 5:18






          $begingroup$
          Ahhh, so the center of the diameter is not the center of the circle, but rather the center of the circle is the center of the diameter
          $endgroup$
          – Dude156
          Feb 11 at 5:18






          1




          1




          $begingroup$
          No, both of those are true. Just added an extra comment in my answer BTW which I think may interest you :)
          $endgroup$
          – David
          Feb 11 at 5:20




          $begingroup$
          No, both of those are true. Just added an extra comment in my answer BTW which I think may interest you :)
          $endgroup$
          – David
          Feb 11 at 5:20












          $begingroup$
          Ah yes, ole' Euclid was a sad guy... He also assumed distributive property... ;)
          $endgroup$
          – Dude156
          Feb 11 at 5:27




          $begingroup$
          Ah yes, ole' Euclid was a sad guy... He also assumed distributive property... ;)
          $endgroup$
          – Dude156
          Feb 11 at 5:27


















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