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### Ellipses: Pythagorean Relationship

```
Date: 2/12/96 at 14:0:31
From: Anonymous
Subject: Ellipses: Pythagorean Relationship

In an ellipse with major axis of 2a, minor axis of 2b, and foci c (on
the major axis), the relationship c squared = a squared - b squared
holds true (often used for finding the foci).  However, on further
examination I was unable to visually understand how the three numbers
fit into a Pythagorean relationship.  C is not a hypotenuse, lying on
the same line as a.

I have consulted my math teacher, and looked in numerous books for an
illustrated explanation, but in vain.

Thanks so much,

Brett
```

```
Date: 9/13/96 at 18:10:8
From: Doctor Ceeks
Subject: Re: Ellipses: Pythagorean Relationship

Hi,

Draw the two lines which connect the foci to one of the points
where the minor axis intersects the ellipse.

You will see two congruent right triangles, back to back.  It's
easy to see that the short sides of these right triangles are b and c.
I claim that the hypotenuse is a.

The reason is that twice that length is the distance from one focus,
to the ellipse, to the other focus, and it is a defining property
of an ellipse that such lengths are constant. So this distance is the
same as going from one focus to a point where the major axis
intersects the ellipse, to the other focus... and that distance, you
can see, is 2a.

-Doctor Ceeks,  The Math Forum
Check out our web site!  http://mathforum.org/dr.math/
```
Associated Topics:
High School Conic Sections/Circles
High School Geometry
High School Triangles and Other Polygons

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