Ellipse reflection property objectively correct proof.

by qwerty123456asdfgzxcvb, Nov 30, 2024, 3:07 AM

Let $\mathcal{C}$ be a (real) conic and let $I,J$ be the circle points, define the foci $F_1,F_2$ as one pair of points made by intersections of tangents from $I,J$ to $\mathcal{C}$ (There are two pairs, pick the pair with both intersection points real).

https://i.ibb.co/wrGpsDf/image.png

Now consider DDIT on the quadrilateral $F_1IF_2J$. There is an involution fixing the tangent at $P$ (call it $\ell$), that swaps $PF_1, PF_2$ and $PI, PJ$. So we have \[(\ell,PF_1; PI, PJ) = (\ell, PF_2; PJ, PI) = \frac{1}{(\ell, PF_2; PI, PJ)}\].
Recall the definition of the angle between two lines $\ell_1, \ell_2$ as $\frac{1}{2i}\ln(\ell_1, \ell_2; \overline{\ell_1 \cap \ell_2 I}, \overline{\ell_1 \cap \ell_2 J})$. Thus, \begin{align*} \angle \ell\overline{PF_1} &= \frac{1}{2i}\ln(\ell, PF_1;PI, PJ) \\ &= \frac{1}{2i}\ln\left( \frac{1}{(\ell, PF_2; PI, PJ)}\right) \\ &= -\frac{1}{2i}\ln(\ell, PF_2;PI, PJ)  = -\angle \ell\overline{PF_2}. \end{align*}
This post has been edited 2 times. Last edited by OronSH, Nov 30, 2024, 3:15 AM

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oh yeah this also proves the fact that for an ellipse with foci F1, F2 and a point P with tangent lines L1, L2, that PF1 and PF2 are isogonal in PL1, PL2 (since the involuion swapping PL1, PL2 swaps PF1, PF2, PI, PJ)

by qwerty123456asdfgzxcvb, Dec 5, 2024, 11:46 PM

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