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My question came from a friend of mine the other day, and I am looking for an answer.

how do you prove the major and minor axis of an ellipse bisect each other?

Thanks

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    $\begingroup$ A formal proof will depend on your definition of an ellipse. Are you thinking of its equation? As a conic section? As the locus of points such that the sum of the distances from the foci is constant? As an affine image of a circle? $\endgroup$ Jun 4 '18 at 20:42
  • $\begingroup$ Standard ellipsis equation. $\endgroup$ Jun 4 '18 at 23:14
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Because an ellipse can always be viewed as a foreshortened circle, the foreshortening does not break symmetry. That is, whatever foreshortening occurs about an axis of rotation through the center of the circle, it always occurs equally either side of that axis (the major axis). The location of the minor axis does not shift from being centered and orthogonal to major axis through that rotation.

This is an explanation rather than a proof. A more formal proof would show how rotation by angle $\alpha$ would reduce the radius of the circle to $r\cdot \cos\alpha$, the minor axis, equally both above and below the axis of rotation while maintaining its center position on the major axis.

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  • $\begingroup$ Can you post the proof? $\endgroup$ Jun 5 '18 at 22:14
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If your ellipse is defined by a standard equation (a quadratic form) then you can translate and rotate the coordinate axes so that the equation becomes $$ \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1. $$ Then the endpoints of the axes are $(\pm a, 0)$ and $(0, \pm b)$ and the origin where they cross clearly bisects both.

Edit in response to comment.

You can read about that rotation here: http://www.stewartcalculus.com/data/ESSENTIAL%20CALCULUS/upfiles/topics/ess_at_13_ra_stu.pdf

You might also get something from

https://en.wikipedia.org/wiki/Ellipse

https://www.maa.org/external_archive/joma/Volume8/Kalman/General.html

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  • $\begingroup$ Can you show this in more formal terms? $\endgroup$ Jun 5 '18 at 22:15
  • $\begingroup$ See my edit for links to more formal proofs. $\endgroup$ Jun 5 '18 at 22:36
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If you are allowed to use affine plane ideas then all ellipses are affinely equivalent to each other and particulary to all circles. In any circle all diameters bisect each other and thus in any ellipse, all line segments passing through the center, including the major and minor axis, bisect each other. A point being in the middle of a line segment is an example of an affine combination of points and thus is affinely invariant.

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  • $\begingroup$ Can you post the proof of this? $\endgroup$ Jun 5 '18 at 22:14
  • $\begingroup$ It is already in MSE question 1498799 "What shape do we get when we shear an ellipse?" $\endgroup$
    – Somos
    Jun 6 '18 at 0:27

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