Jtdylktuy

I think I have found out some property of an ellipse. Define the following:

• The point $P$ belongs to the ellipse.


• The tangent line $ell$ goes through $P$.


• The point $F$ is one of the foci.


• The line $m$, through $F$, is perpendicular to the line $ell$ at point $I$.


• 
  $A$ and $B$ are the vertices of the ellipse (that is, the endpoints of the major axis).

In this case,

The $angle AIB$ is always $pi/2$. (I think.....)

But I didn't prove yet...

1. Is it true?


2. Could anybody give me some advice?

Thanks in advance.

Edit1 For jmerry

Edit2 For jmerry

Really thanks, makes me fun!

Yes, it's true.

Some advice: the locus of all points $I$ such that $angle AIB$ is a right angle is the circle with diameter $AB$. So, then, if you can show that $OI=OA=OB$ where $O$ is the center of the ellipse, you'll have it.

Next, add some more things to your diagram. The other focus (let's call it $G$) looks like a good place to start - after all, that lets us use nice facts like the segments from $P$ to the two focuses making the same angle to the tangent line $ell$, and that $PF+PG=AB$.

So now, how can you use those facts? Don't be afraid to extend lines and find new intersections.

[Added in edit]

All right, you've drawn some more things in, and some equal angles. Since we're looking at $I$ in particular, how about naming that point where $GP$ and $FI$ intersect? You've already labeled an angle there, so naming the point will make it easier to work with.

(I called it $J$ in my diagram)

Oh, and since we're interested in the length of $OI$, that's another segment we should draw.

Hint: The only way for the angle $AIB$ to always be $frac{pi}{2}$ is if the point $I$ lies along a circle with radius $|B|$ (Check out Thales's Theorem). In other words, the distance from I to the origin must be constant.

Not a strict proof, an interesting observation though.

There is a method to draw an ellipse by folding paper. See this video: Folding a Circle into an Ellipse. In case the link dies, of if you want more information, search for folding paper ellipse or so.

The method goes like this:

1. Draw a circle on a piece of paper, $G$ is the center.


2. Choose a point $F$ inside the circle.


3. Repeat many times ($n=0,1,2,3,…$) the following:
   
   
   
   
   1. choose a point $J_n$ on the circle;
   
   
   2. fold the paper so $J_n$ meets $F$;
   
   
   3. unfold, denote the folding line $l_n$.
   
   
   
   


4. An ellipse will appear with $F$ and $G$ as foci; each $l_n$ will be some tangent line.

In the question you're working in reverse! You have the ellipse, you have $F$ and $G$. You choose some $l$ and find corresponding $J$. All possible $J$-like points should then form a circle which I denote $mathcal{J}$; we expect $G$ to be its center.

How about $I$ and all possible $I$-like points (the set I denote $mathcal{I}$)? From the fact that $overrightarrow{FI} = frac 1 2 overrightarrow{FJ}$ we conclude $mathcal{I}$ should be a homogeneous dilation of $mathcal{J}$ with the center $F$ and the ratio $frac 1 2$. So all possible $I$-like points also form a circle. Applying the dilation to the center of the circle $mathcal{J}$ (i.e. to the point $G$) we obtain the center of the circle $mathcal{I}$. In your case it's the origin $O$.

Now consider tangent lines at $A$ and at $B$. It's easy to see these two points must belong to $mathcal{I}$. Knowing $O$ is the center of $mathcal{I}$, we conclude $AB$ is the diameter.

Your $I$ is on the circle $mathcal{I}$, the diameter is $AB$. Therefore $angle AIB = frac pi 2$.