Plane Geometry Procedure?

Question

I have a right quadrilateral (rectangle) of sides A and B.

I need to find the foci of an ellipse that will fit tangent to the center points of all four sides of the rectangle, using nothing but a compass and straight edge.

"Quickie Diagram"
http://i122.photobucket.com/albums/o248/Web_Eagle/fociproblem.gif

I suspect the answer will involve bisecting the sides, setting the compass equal to the diagonal of one of the quadrants, and then marking the longer center line from opposite edges.

BUT, I’m not positive, and on this particular project, I only get one shot. (This actually involves construction, where I will be drilling and cutting into a non-reproducible surface.)

Can anybody tell me clearly if I’m correct, or how else to find these foci? Best proof gets the points!

Thanks!

Answer 1

an ellipse has c² = a² - b²
where c = focus, a = major axis, b = minor axis
Here you have a = B/2 and b = A/2
B is the length and A is the height of the rectangle.

The center of the ellipse is the intersection of two
perpendicular bisectors.
a² = b² + c² and a is the hypotenuse of a rt. triangle.

You just need to construct a rt. triangle with a leg and a
hypotenuse, then you can find the leg c, which is the focus,
the distance from the center of the rectangle.

Bisect side A and side B, you will get length b and a.

To find c, draw a right triangle inscribed in a semi- circle.

Draw a circle with diameter MN with length a.
Construct a chord MP of length b where P is a point on the
circle. MNP forms a right triangle with hypotenuse a and
leg b. Chord NP is c, which is the distance of the focus.

Answer 2

You're starting out right.

1) Bisect the sides of the rectangle. Then draw major and minor axes. They will intersect in the center of the rectangle and divide it into four identical parts. On the horizontal axis label the left edge A, the center O, and the right hand edge A'. Label the upper left hand corner of the rectangle B.

2) Bisect the left semi-major axis (the line segment AO). Call the center point of that line segment M.

3) Construct the circle with center M and radius MO. Notice that MA is also a radius. Now construct a circle with center A and radius AB.

4) Circles M and A intersect in the upper left hand quadrant at point P. Ignore the other point of intersection in the lower left hand quadrant.

5) The line segment OP has the focal length c. Now construct a circle with center O and radius OP. It will intersect the major axis AA' at the two focal points.