I found the following nice problem in my Facebook account. Facebook, however, is a miracle to me and I am always unable to find a posting a second time. Unless you answer something silly like „Following“ it is lost. The best I could find was this link.

The problem is to prove:

The intersections of two perpendicular tangents to an ellipse form a circle.

Of course, this can be computed by Analytic Geometry and I carry this out below. It is no fun, however. E.g., you can find the radius of the circle if the claim is correct, take a point on the circle, compute the two tangents to the ellipse and check that they are perpendicular.

Let us find a more geometrical solution! At first, there is a well known „construction“ of ellipses folding a circular paper. You can do it with actual paper. Cut a circular paper. Mark a point A inside the circle. Then fold the paper, so that the boundary lies on A. I.e., the point P is reflected along the folding line to A. The line is the middle perpendicular of AP. If you do that often enough the folds will outline the green ellipse. There are even videos on Youtube showing this.

For the proof, you need that ellipses are the set of points where the sum of distances to A and M is constant. In our case, it is „obviously“ constantly equal to the radius of the circle. We have the following.

The set of all middle perpendiculars to a fixed point A and points P on a circle is the set of tangents to an ellipse.

Now you know how to get two perpendicular tangents. In the following construction, I did just that. The four blue lines are four middle perpendiculars. They are constructed by selecting a point P on the black circle, the line PA and a rectangular to that line. Then the blue lines are the four middle perpendiculars on A and the four intersections of our green lines with the circle.

We have to prove that

• the midpoint Z on AM is the center of the rectangle
• and that the rectangle has the same diameter, independent on the choice of P on the black circle.

Then the corners of the blue rectangle will always be on the same circle.

Now, we stretch the blue rectangle by a factor 2 from the center at A. The point Z will be stretched to the point M then. The blue rectangle will become a rectangle through the intersections as in the following construction.

It is now „quite obvious“ that M is the center of the blue rectangle. Thus Z was the center of the 1/2 times smaller rectangle. This proves our first claim.

For the second claim, we need to show that the length of the diagonal of the large blue rectangle does not depend on the choice of the red point. The diagonal has the length

\(\sqrt{c^2+d^2}\)

by Pythagoras. Now we have another claim.

The sum of the squares of the lengths of each two perpendicular secants of a fixed circle that meet in a fixed point inside the circle is constant.

To prove that have a look at the dashed rectangle with diagonal AM. Using this rectangle and Pythagoras you can „easily“ express the diagonal of the blue rectangle by the length of AM and the radius of the circle. This proves the second claim.

The images in this posting have been done with C.a.R. (Compass and Ruler), a Java program developed by the author. It allows beautiful exports of images and the automatic creation of polar sets for sets of lines. That feature was used to compute the ellipse in the second image. The first ellipse was done using the two focal points. C.a.R. has also ellipses defined by 5 points, or even by an equation or a parameterization.

I promised to compute the problem using Analytic Geometry. I am using the Computer Algebra system Maxima via my Euler Math Toolbox for this.

The first method computes two perpendicular tangents to the ellipse with the equation

\(x^2 + c^2 y^2 = 1\)

To find a tangent perpendicular to a vector v, we can maximize the expression

\(v_1 x + v_2 y\)

on the ellipse using the method of Lagrange. If the vector v has norm 1 the value of this maximum will be the distance of the tangent from the origin. The Lagrange equations fot his are

\(v_1 = 2 \lambda x, \quad v_2 = 2 \lambda c^2 y, \quad x^2+c^2 y^2 = 1\)

After solving this, we get

\(v_1 x + v_2 y = 2  \lambda (x^2+c^2y^2) = 2 \lambda\)

Then we do the same with the orthogonal vector, i.e., we maximize

\(-v_2 x + v_1 y\)

We then show that the sum of squares of these two values is constant.

In EMT and Maxima, this is the following code.

>&solve([v1=2*la*x,v2=2*la*c^2*y,x^2+c^2*y^2=1],[x,y,la]); ...
>  la1 &= la with %[1]
 
                                  2    2   2
                           sqrt(v2  + c  v1 )
                           ------------------
                                  2 c
 
>&solve([-v2=2*la*x,v1=2*la*c^2*y,x^2+c^2*y^2=1],[x,y,la]);  ...
>  la2 &= la with %[1]
 
                                 2   2     2
                           sqrt(c  v2  + v1 )
                           ------------------
                                  2 c
 
>&factor(la1^2+la2^2)
 
                            2         2     2
                          (c  + 1) (v2  + v1 )
                          --------------------
                                     2
                                  4 c

Thus the circle of intersections has the radius

\(r = \dfrac{\sqrt{1+c^2}}{c} = \sqrt{1 + \dfrac{1}{c^2}}\)

This is confirmed by the special case of tangents parallel to the axes.

There are several other methods. One is to construct the tangents using tangents to a circle. For this, the ellipse needs to be stretched by 1/c into y-direction. It will then become a circle. We need to compute points on the image of our circle with radius r under this mapping, then take the tangents to the mapped ellipse and map back. Below is the plan of such a construction.

The computations are very involved, however.

Another method to find both tangents is the following: We look at all lines through a given point (e.g. determined by their slope) and find the ones that intersect the ellipse only once. The product of the two slopes that solve this problem should be 1. Again, this is a complicated computation.