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Construct Ellipse Axes (Posted on 2004-12-21) Difficulty: 4 of 5
Given an arbitrary ellipse (not a circle) without a marked center or foci. Using a straightedge and compass construct the major and minor axes.

See The Solution Submitted by Bractals    
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Unproven/Incomplete solution | Comment 2 of 4 |

1)  Pick any point O, on the ellipse

2)  Construct circle O which intersects the ellipse in four points (A, B, C, D in that order)  (I assume this can alway be done.)

3)  Construct segments AC and BD.  Call the intersection of these two segments Z.

4)  Constuct the bisectors of the angles AZB, BZC, CZD, and DZA.  (These bisectors are parallel to the axes of the ellipse, but I haven't proven this.)

5)  Construct a tangent to the ellipse at point A.  Call this line l (I don't know if this can be done.)

6)  Construct another tangent to the ellipse which is parallel to l.  Call the point of tangency E.  (I don't know if this can be done.)

7)  Construct segment AE.  This line passes through the center of the ellipse.  (Unproven)

8)  Repeat steps 5-7 using point B and other point of tangency F.

9)  Call the intersection of AE and BF point P.  This is the center of the ellipse.

10)  Translate the four angle bisectors from step 4 along the vector from Z to P.

I *think* that works, but it is far from a complete solution.

-Jer


  Posted by Jer on 2004-12-21 18:08:51
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