Ellipse reflection property objectively correct proof.
by qwerty123456asdfgzxcvb, Nov 30, 2024, 3:07 AM
Let 
be a (real) conic and let 
be the circle points, define the foci 
as one pair of points made by intersections of tangents from to (There are two pairs, pick the pair with both intersection points real).
Now consider DDIT on the quadrilateral
. There is an involution fixing the tangent at 
(call it 
), that swaps 
and 
. So we have ![\[(\ell,PF_1; PI, PJ) = (\ell, PF_2; PJ, PI) = \frac{1}{(\ell, PF_2; PI, PJ)}\]](//latex.artofproblemsolving.com/7/a/7/7a7733c367fe097c9a5cb01b6465f4797eb9cb65.png)
.
Recall the definition of the angle between two lines
as 
. Thus, 
be a (real) conic and let 
be the circle points, define the foci 
as one pair of points made by intersections of tangents from to (There are two pairs, pick the pair with both intersection points real).Now consider DDIT on the quadrilateral
. There is an involution fixing the tangent at 
(call it 
), that swaps 
and 
. So we have ![\[(\ell,PF_1; PI, PJ) = (\ell, PF_2; PJ, PI) = \frac{1}{(\ell, PF_2; PI, PJ)}\]](http://latex.artofproblemsolving.com/7/a/7/7a7733c367fe097c9a5cb01b6465f4797eb9cb65.png)
.Recall the definition of the angle between two lines
as 
. Thus, 
This post has been edited 2 times. Last edited by OronSH, Nov 30, 2024, 3:15 AM
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